Approximation of the Cut Function by Some Generic Logistic Functions and Applications
Advances in Applied Sciences
Volume 1, Issue 2, October 2016, Pages: 24-29
Received: Aug. 17, 2016;
Accepted: Aug. 27, 2016;
Published: Sep. 12, 2016
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Nikolay Kyurkchiev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
Svetoslav Markov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
In this paper we study the uniform approximation of the cut function by smooth sigmoid functions such as Nelder and Turner–Blumenstein–Sebaugh growth functions. To illustrate the use of one of the models we have fitted the model to the “classical Verhulst data”. Several numerical examples are presented throughout the paper using the contemporary computer algebra system MATHEMATICA.
Approximation of the Cut Function by Some Generic Logistic Functions and Applications, Advances in Applied Sciences.
Vol. 1, No. 2,
2016, pp. 24-29.
S. Shoffner and S. Schnell, Estimation of the lag time in a subsequent monomer addition model for fibril elongation, bioRxiv The preprint server for biology, 2015, pp. 1–8, doi: 10.1101/034900.
P. Arosio, T. P. J. Knowles, and S. Linse, On the lag phase in amyloid fibril formation, Physical Chemistry Chemical Physics, vol. 17, 2015, pp. 7606–7618, doi: 10.1039/C4CP05563B.
N. Kyurkchiev, A note on the new geometric representation for the parameters in the fibril elongation process, Compt. rend. Acad. bulg. Sci., vol. 69 (8), 2016, pp. 963–972.
S. Markov, Building reaction kinetic models for amyloid fibril growth, BIOMATH, vol. 5, 2016, http://dx.dpi.org/10.11145/j.biomath.2016.07.311.
M. Turner, B. Blumenstein, and J. Sebaugh, A Generalization of the Logistic Law of Growth, Biometrics, vol. 25 (3), 1969, pp. 577–580.
J. A. Nelder, The fitting of a generalization of the logistic curve, Biometrics, vol. 17, 1961, pp. 89–110.
P. F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance mathematique et physique, vol. 10, 1838, pp. 113–121.
N. Kyurkchiev and S. Markov, On the Hausdorff distance between the Heaviside step function and Verhulst logistic function, J. Math. Chem., vol. 54 (1), 2016, pp. 109–119, doi: 10.1007/S10910-015-0552-0.
N. Kyurkchiev and S. Markov, Sigmoidal functions: some computational and modelling aspects, Biomath Communications, vol. 1 (2), 2014, pp. 30–48, doi: 10.11145/j.bmc.2015.03.081.
A. Iliev, N. Kyurkchiev and S. Markov, On the approximation of the cut and step functions by logistic and Gompertz functions, Biomath, vol. 4 (2), 2015, pp. 2–13.
N. Kyurkchiev and S. Markov, On the approximation of the generalized cut function of degree by smooth sigmoid functions, Serdica J. Computing, vol. 9 (1), 2015, pp. 101–112.
A. Iliev, N. Kyurkchiev, and S. Markov, On the Approximation of the step function by some sigmoid functions, Mathematics and Computers in Simulation, 2015, doi: 10.1016/j.matcom.2015.11.005.
N. Kyurkchiev and A. Iliev, On some growth curve modeling: approximation theory and applications, Int. J. of Trends in Research and Development, vol. 3 (3), 2016, pp. 466–471, http://www.ijtrd.com/papers/IJTRD3869.pdf
N. Kyurkchiev and S. Markov, Sigmoid functions: Some Approximation and Modelling Aspects, LAP LAMBERT Academic Publishing, Saarbrucken, 2015, ISBN 978-3-659-76045-7.
N. Kyurkchiev and A. Iliev, A note on some growth curves arising from Box-Cox transformation, Int. J. of Engineering Works, vol. 3 (6), 2016, pp. 47–51, ISSN: 2409-2770.
N. Kyurkchiev, S. Markov, and A. Iliev, A note on the Schnute growth model, Int. J. of Engineering Research and Development, vol. 12 (6), 2016, pp. 47–54, ISSN: 2278-067X, http://www.ijerd.com/paper/vol12-issue6/Verison-1/G12614754.pdf
A. Iliev, N. Kyurkchiev, and S. Markov, On the Hausdorff distance between the shifted Heaviside step function and the transmuted Stannard growth function, BIOMATH, 2016, (accepted).
N. Kyurkchiev, On the Approximation of the step function by some cumulative distribution functions, Compt. rend. Acad. bulg. Sci., vol. 68 (12), 2015, pp. 1475–1482.
V. Kyurkchiev and N. Kyurkchiev, On the Approximation of the Step function by Raised-Cosine and Laplace Cumulative Distribution Functions, European International Journal of Science and Technology, vol. 4 (9), 2016, pp. 75–84.
D. Costarelli and R. Spigler, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks, vol. 44, 2013, pp. 101–106.
D. Costarelli and G. Vinti, Pointwise and uniform approximation by multivariate neural network operators of the max-product type, Neural Networks, 2016, doi: 10.1016/j.neunet.2016.06.002.
Costarelli, D., R. Spigler, Solving numerically nonlinear systems of balance laws by multivariate sigmoidal functions approximation, Computational and Applied Mathematics, 2016, doi: 10.1007/s40314-016-0334-8.
D. Costarelli and G. Vinti, Convergence for a family of neural network operators in Orlicz spaces, Mathematische Nachrichten, 2016, doi: 10.1002/mana.20160006.
N. Guliyev and V. Ismailov, A single hidden layer feedforward network with only one neuron in the hidden layer san approximate any univariate function, Neural Computation, vol. 28, 2016, pp. 1289–1304.
J. Dombi and Z. Gera, The Approximation of Piecewise Linear Membership Functions and Lukasiewicz Operators, Fuzzy Sets and Systems, vol. 154 (2), 2005, pp. 275–286.