A new class activation functions with application in the theory of impulse technics

Nikolay Kyurkchiev

doi:10.33187/jmsm.421392

Research Article

A new class activation functions with application in the theory of impulse technics

Year 2018, Volume: 1 Issue: 1, 15 - 20, 27.05.2018

Nikolay Kyurkchiev

https://doi.org/10.33187/jmsm.421392

Abstract

In this note we define the new activation functions, based on the well-known hyperbolic tangent and half--hyperbolic tangent activation functions. We consider the Hausdorff distance between the ''double step'' function $\sigma^{\ast}(t)$ (resp. function $\sigma^{\ast \ast}(t)$) and the new classes of activation functions. The results have independent significance in the study of issues related to neural networks and impulse techniques. Numerical examples, illustrating our results are presented using programming environment Mathematica.

Keywords

''double step'' function $\sigma^{\ast}(t)$, $\sigma^{\ast \ast}(t)$--function, emitting chart, Hausdorff distance

References

[1] F. HAUSDORFF, Set Theory (2 ed.), New York, Chelsea Publ., 1962.
[2] B. SENDOV, Hausdorff Approximations Boston, Kluwer, 1990.
[3] N. KYURKCHIEV AND A. ANDREEV, Approximation and antenna and filter synthesis: Some moduli in programming environment Mathematica, LAP LAMBERT Academic Publishing, Saarbrucken, 2014; ISBN 978-3-659-53322-8.
[4] N. KYURKCHIEV AND BL. SENDOV, Approximation of a class of functions by algebraic polynomials with respect to Hausdorff distance, Ann. Univ. Sofia, Fac. Math., vol. 67, pp 573–579, 1975 (in Bulgarian).
[5] N. KYURKCHIEV AND S. MARKOV, On the Hausdorff distance between the Heaviside step function and Verhulst logistic function, J. Math. Chem., vol. 54, no 1, pp 109–119, 2016.
[6] A. ANDREEV AND N. KYURKCHIEV, Approximation of some impulse functions - implementation in programming environment MATHEMATICA, Proceedings of the 43 Spring Conference of the Union of Bulgarian Mathematicians, Borovetz, April 2-6, 2014, pp 111-117.
[7] N. KYURKCHIEV AND S. MARKOV, On the numerical approximation of the ”cross” set, Ann. Univ. Sofia, Fac. Math., Vol. 66, pp 19–25, 1974 (in Bulgarian).
[8] N. KYURKCHIEV AND A. ANDREEV, Hausdorff approximation of functions different from zero at one point - implementation in programming environment MATHEMATICA, Serdica J. of Computing, vol. 7, no 2, pp 135–142, 2013.
[9] N. KYURKCHIEV AND A. ANDREEV, Synthesis of slot aerial grids with Hausdorff–type directive patterns – implementation in programming environment Mathematica, C.R. Acad. Bulgare Sci., vol. 66, No 11, pp 1521–1528, 2013.
[10] N. KYURKCHIEV, Synthesis of slot aerial grids with Hausdorff type directive patterns, PhD Thesis, Department of Radio-Electronics, VMEI, Sofia, 1979 (in Bulgarian).
[11] BL. SENDOV, H. SCHINEV AND N. KJURKCHIEV, Hausdorff-synthesis of aerial grids in scanning the directive diagram, Electropromishlenost i Priboroostroene, vol. 16, no 5, pp 203–205, 1981 (in Bulgarian).
[12] H. SCHINEV, N. KJURKCHIEV AND M. GACHEV, Experimental investigations of slot aerial grids with Hausdorff type directive patterns, Electropromishlenost i Priboroostroene, vol. 14, no 6, pp. 223–224, 1979 (in Bulgarian).
[13] H. SHINEV, N. KYURKCHIEV, M. GACHEV AND S. MARKOV, Application of a class of polynomials of best approximation to linear antenna array synthesis, Izv. VMEI, Sofia, vol. 34, no 1, pp. 1–6, 1975 (in Bulgarian).
[14] A. GOLEV, T. DJAMIYKOV AND N. KYURKCHIEV, Sigmoidal functions in antenna-feeder technique, Int. J. of Pure and Appl. Math., vol. 116, no 4, pp 1081–1092, 2017.
[15] N. KYURKCHIEV, A. ILIEV AND S. MARKOV, Some techniques for recurrence generating of activation functions, LAP LAMBERT Academic Publishing, 2017; ISBN 978-3-330-33143-3
[16] V. KYURKCHIEV AND N. KYURKCHIEV, A family of recurrence generated functions based on Half-hyperbolic tangent activation functions, Biomedical Statistics and Informatics, vol. 2, no 3, pp 87–94, 2017.
[17] 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, pp 1289-–1304, 2016.
[18] D. COSTARELLI AND R. SPIGLER, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks, vol. 44, pp 101-–106, 2013.
[19] 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
[20] D. COSTARELLI AND 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
[21] D. COSTARELLI AND G. VINTI, Convergence for a family of neural network operators in Orlicz spaces, Mathematische Nachrichten, 2016; doi: 10.1002/mana.20160006

Year 2018, Volume: 1 Issue: 1, 15 - 20, 27.05.2018

Nikolay Kyurkchiev

https://doi.org/10.33187/jmsm.421392

Abstract

References

[1] F. HAUSDORFF, Set Theory (2 ed.), New York, Chelsea Publ., 1962.
[2] B. SENDOV, Hausdorff Approximations Boston, Kluwer, 1990.
[3] N. KYURKCHIEV AND A. ANDREEV, Approximation and antenna and filter synthesis: Some moduli in programming environment Mathematica, LAP LAMBERT Academic Publishing, Saarbrucken, 2014; ISBN 978-3-659-53322-8.
[4] N. KYURKCHIEV AND BL. SENDOV, Approximation of a class of functions by algebraic polynomials with respect to Hausdorff distance, Ann. Univ. Sofia, Fac. Math., vol. 67, pp 573–579, 1975 (in Bulgarian).
[5] N. KYURKCHIEV AND S. MARKOV, On the Hausdorff distance between the Heaviside step function and Verhulst logistic function, J. Math. Chem., vol. 54, no 1, pp 109–119, 2016.
[6] A. ANDREEV AND N. KYURKCHIEV, Approximation of some impulse functions - implementation in programming environment MATHEMATICA, Proceedings of the 43 Spring Conference of the Union of Bulgarian Mathematicians, Borovetz, April 2-6, 2014, pp 111-117.
[7] N. KYURKCHIEV AND S. MARKOV, On the numerical approximation of the ”cross” set, Ann. Univ. Sofia, Fac. Math., Vol. 66, pp 19–25, 1974 (in Bulgarian).
[8] N. KYURKCHIEV AND A. ANDREEV, Hausdorff approximation of functions different from zero at one point - implementation in programming environment MATHEMATICA, Serdica J. of Computing, vol. 7, no 2, pp 135–142, 2013.
[9] N. KYURKCHIEV AND A. ANDREEV, Synthesis of slot aerial grids with Hausdorff–type directive patterns – implementation in programming environment Mathematica, C.R. Acad. Bulgare Sci., vol. 66, No 11, pp 1521–1528, 2013.
[10] N. KYURKCHIEV, Synthesis of slot aerial grids with Hausdorff type directive patterns, PhD Thesis, Department of Radio-Electronics, VMEI, Sofia, 1979 (in Bulgarian).
[11] BL. SENDOV, H. SCHINEV AND N. KJURKCHIEV, Hausdorff-synthesis of aerial grids in scanning the directive diagram, Electropromishlenost i Priboroostroene, vol. 16, no 5, pp 203–205, 1981 (in Bulgarian).
[12] H. SCHINEV, N. KJURKCHIEV AND M. GACHEV, Experimental investigations of slot aerial grids with Hausdorff type directive patterns, Electropromishlenost i Priboroostroene, vol. 14, no 6, pp. 223–224, 1979 (in Bulgarian).
[13] H. SHINEV, N. KYURKCHIEV, M. GACHEV AND S. MARKOV, Application of a class of polynomials of best approximation to linear antenna array synthesis, Izv. VMEI, Sofia, vol. 34, no 1, pp. 1–6, 1975 (in Bulgarian).
[14] A. GOLEV, T. DJAMIYKOV AND N. KYURKCHIEV, Sigmoidal functions in antenna-feeder technique, Int. J. of Pure and Appl. Math., vol. 116, no 4, pp 1081–1092, 2017.
[15] N. KYURKCHIEV, A. ILIEV AND S. MARKOV, Some techniques for recurrence generating of activation functions, LAP LAMBERT Academic Publishing, 2017; ISBN 978-3-330-33143-3
[16] V. KYURKCHIEV AND N. KYURKCHIEV, A family of recurrence generated functions based on Half-hyperbolic tangent activation functions, Biomedical Statistics and Informatics, vol. 2, no 3, pp 87–94, 2017.
[17] 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, pp 1289-–1304, 2016.
[18] D. COSTARELLI AND R. SPIGLER, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks, vol. 44, pp 101-–106, 2013.
[19] 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
[20] D. COSTARELLI AND 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
[21] D. COSTARELLI AND G. VINTI, Convergence for a family of neural network operators in Orlicz spaces, Mathematische Nachrichten, 2016; doi: 10.1002/mana.20160006

There are 21 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Nikolay Kyurkchiev
Publication Date	May 27, 2018
Submission Date	May 6, 2018
Acceptance Date	May 14, 2018
Published in Issue	Year 2018 Volume: 1 Issue: 1

Cite

APA	Kyurkchiev, N. (2018). A new class activation functions with application in the theory of impulse technics. Journal of Mathematical Sciences and Modelling, 1(1), 15-20. https://doi.org/10.33187/jmsm.421392
AMA	Kyurkchiev N. A new class activation functions with application in the theory of impulse technics. Journal of Mathematical Sciences and Modelling. May 2018;1(1):15-20. doi:10.33187/jmsm.421392
Chicago	Kyurkchiev, Nikolay. “A New Class Activation Functions With Application in the Theory of Impulse Technics”. Journal of Mathematical Sciences and Modelling 1, no. 1 (May 2018): 15-20. https://doi.org/10.33187/jmsm.421392.
EndNote	Kyurkchiev N (May 1, 2018) A new class activation functions with application in the theory of impulse technics. Journal of Mathematical Sciences and Modelling 1 1 15–20.
IEEE	N. Kyurkchiev, “A new class activation functions with application in the theory of impulse technics”, Journal of Mathematical Sciences and Modelling, vol. 1, no. 1, pp. 15–20, 2018, doi: 10.33187/jmsm.421392.
ISNAD	Kyurkchiev, Nikolay. “A New Class Activation Functions With Application in the Theory of Impulse Technics”. Journal of Mathematical Sciences and Modelling 1/1 (May 2018), 15-20. https://doi.org/10.33187/jmsm.421392.
JAMA	Kyurkchiev N. A new class activation functions with application in the theory of impulse technics. Journal of Mathematical Sciences and Modelling. 2018;1:15–20.
MLA	Kyurkchiev, Nikolay. “A New Class Activation Functions With Application in the Theory of Impulse Technics”. Journal of Mathematical Sciences and Modelling, vol. 1, no. 1, 2018, pp. 15-20, doi:10.33187/jmsm.421392.
Vancouver	Kyurkchiev N. A new class activation functions with application in the theory of impulse technics. Journal of Mathematical Sciences and Modelling. 2018;1(1):15-20.