On the Hausdorff distance between the Heaviside step function and Verhulst logistic function

2015 ◽  
Vol 54 (1) ◽  
pp. 109-119 ◽  
Author(s):  
Nikolay Kyurkchiev ◽  
Svetoslav Markov
Keyword(s):  
Hausdorff Distance ◽  
Logistic Function ◽  
Step Function ◽  
Heaviside Step Function

scholarly journals On the Hausdorff distance between the shifted Heaviside step function and the transmuted Stannard growth function

BIOMATH ◽  
2016 ◽  
Vol 5 (2) ◽  
pp. 1609041 ◽  
Author(s):  
Anton Iliev Iliev ◽  
Nikolay Kyurkchiev ◽  
Svetoslav Markov
Keyword(s):  
Hausdorff Distance ◽  
Growth Curves ◽  
Growth Function ◽  
Step Function ◽  
Upper And Lower Bounds ◽  
Programming Environment ◽  
Software Module ◽  
Heaviside Step Function ◽  
Numerical Examples ◽  
The One

In this paper we study the one-sided Hausdorff distance between the shifted Heaviside step--function and the transmuted Stannard growth function. Precise upper and lower bounds for the Hausdorff distance have been obtained. We present a software module (intellectual property) within the programming environment CAS Mathematica for the analysis of the growth curves. Numerical examples, illustrating our results are given, too.

Dynamics of a Tubular Cantilever Simultaneously Subjected to Internal and Partially-Confined External Axial Flows

2018 ◽  
Author(s):  
Ahmed R. Abdelbaki ◽  
Arun K. Misra ◽  
Michael P. Païdoussis
Keyword(s):  
Logistic Function ◽  
Step Function ◽  
Dynamical Behaviour ◽  
External Flow ◽  
Annular Region ◽  
Heaviside Step Function ◽  
Cantilever Length ◽  
Theoretical Predictions ◽  
External Flows ◽  
The Stability

In this paper the dynamics of a tubular cantilever, simultaneously subjected to internal and external axial flows, is examined theoretically. The tube is discharging fluid downwards which then flows upwards through an annular region surrounding the tube. Thus, the internal and external flows are interdependent and in opposite directions. Also, the external flow is confined over a certain range of the cantilever length and unconfined over the rest. The Heaviside step function has been used in the literature, for such a system, to model the discontinuity in the external flow velocity occurring when the flow enters the annular region. A more accurate way to model this discontinuity is introduced in this study, in which the logistic function is used instead of the Heaviside step function. The stability of the system is investigated by analysis of the system eigenfrequencies, and the effects of varying the length of the confined region are theoretically studied. The obtained results are compared to theoretical predictions and experimental data from the literature having the same system parameters. The proposed theory captures the same dynamical behaviour as observed experimentally, and has a better estimation for the onset of instability and the frequency of oscillations compared to the theory in the literature.

A family of recurrence generated sigmoidal functions based on the Verhulst logistic function. Some approximation and modelling aspects

2016 ◽  
Vol 3 (2) ◽  
Author(s):  
Nikolay V. Kyurkchiev
Keyword(s):  
Logistic Function ◽  
Step Function ◽  
Heaviside Step Function ◽  
Numerical Examples ◽  
Sigmoidal Functions ◽  
Hausdorff Approximation ◽  
Logistic Functions

  In this note we construct a family of recurrence generated sigmoidal logistic functions based on the Verhulst logistic function.We prove estimates for the Hausdorff approximation of the Heaviside step function by means of this family. Numerical examples, illustrating our results are given.

scholarly journals A family of recurrence generated sigmoidal functions based on the Verhulst logistic function. Some approximation and modelling aspects

2016 ◽  
Vol 3 (2) ◽  
Author(s):  
Nikolay Kyurkchiev
Keyword(s):  
Logistic Function ◽  
Step Function ◽  
Heaviside Step Function ◽  
Numerical Examples ◽  
Sigmoidal Functions ◽  
Hausdorff Approximation ◽  
Logistic Functions

In this note we construct a family of recurrence generated sigmoidal logistic functions based on the Verhulst logistic function.We prove estimates for the Hausdorff approximation of the Heaviside step function by means of this family. Numerical examples, illustrating our results are given.

scholarly journals A family of recurrence generated sigmoidal functions based on the Log-logistic function. Some approximation aspects

2018 ◽  
Vol 5 (1) ◽  
Author(s):  
Nikolay Kyurkchiev ◽  
Svetoslav Markov
Keyword(s):  
Logistic Function ◽  
Step Function ◽  
Biochemical Reactions ◽  
Heaviside Step Function ◽  
Numerical Examples ◽  
Sigmoidal Functions ◽  
Hausdorff Approximation

In this note we construct a family of recurrence generated sigmoidal functions based on the Log--logistic function. The study of some biochemical reactions is linked to a precise Log--logistic function analysis.We prove estimates for the Hausdorff approximation of the Heaviside step function by means of this family. Numerical examples, illustrating our results are given. The plots are prepared using CAS Mathematica.

scholarly journals A new transmuted cumulative distribution function based on the Verhulst logistic function with application in population dynamics

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Nikolay Kyurkchiev
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Logistic Function ◽  
Step Function ◽  
Cumulative Distribution ◽  
Heaviside Step Function ◽  
Numerical Examples ◽  
New Class ◽  
Sigmoidal Functions ◽  
Hausdorff Approximation

In this note we find application of a new class cumulative distribution function transformations to construct a family of sigmoidal functions based on the Verhulst logistic function. We prove estimates for the Hausdorff approximation of the shifted Heaviside step function by means of this family. Numerical examples, illustrating our results are given.

A Meshless Local Petrov-Garlerkin Method for Solving the Biharmonic Equation

2014 ◽  
Vol 931-932 ◽  
pp. 1488-1494
Author(s):  
Supanut Kaewumpai ◽  
Suwon Tangmanee ◽  
Anirut Luadsong
Keyword(s):  
Relative Error ◽  
Biharmonic Equation ◽  
Interpolation Method ◽  
Step Function ◽  
Special Treatment ◽  
Maximum Relative Error ◽  
Heaviside Step Function ◽  
Treatment Techniques ◽  
Point Interpolation Method ◽  
Point Interpolation

A meshless local Petrov-Galerkin method (MLPG) using Heaviside step function as a test function for solving the biharmonic equation with subjected to boundary of the second kind is presented in this paper. Nodal shape function is constructed by the radial point interpolation method (RPIM) which holds the Kroneckers delta property. Two-field variables local weak forms are used in order to decompose the biharmonic equation into a couple of Poisson equations as well as impose straightforward boundary of the second kind, and no special treatment techniques are required. Selected engineering numerical examples using conventional nodal arrangement as well as polynomial basis choices are considered to demonstrate the applicability, the easiness, and the accuracy of the proposed method. This robust method gives quite accurate numerical results, implementing by maximum relative error and root mean square relative error.

The Mesoscopic Investigation on Moduli Prediction in Visco-Elastic Particle Reinforced Composites

2008 ◽  
Vol 385-387 ◽  
pp. 329-332
Author(s):  
Xue Zhong Ding ◽  
Li Qiang Tang
Keyword(s):  
Volume Fraction ◽  
Step Function ◽  
Elastic Behavior ◽  
Loading Path ◽  
Elastic Model ◽  
Reinforced Composites ◽  
Heaviside Step Function ◽  
Equivalent Inclusion ◽  
Inclusion Theory ◽  
Elastic Particle

The visco-elastic mechanism of particles reinforced composites has been investigated through revised Eshelby equivalent inclusion theory. A visco-elastic model is applied. Furthermore, by introducing Heaviside step function and Laplace transform, the creep constitutional equation related to strain rate effect is achieved. Finally, by equivalent inclusion theory, introducing secant modulus, the material moduli with time and volume fraction concerning Glass/ED6 particles reinforced materials have been given. The results show that the visco-elastic property of composite material is mainly determined by the visco-elastic behavior of the matrix, which meet experiment results well. It can be concluded from the results that there exits close relationship between the inclusion shape, volume fraction and loading path.

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