Abstract
Division by zero (DBZ) problem, or say, division singularity problem, has perplexed scientists and engineers in many fields for centuries. How to solve DBZ problem has actually been discussed for more than 1200 years. Despondingly, plenty of efforts failed to solve DBZ problem effectively, and it is still considered as a formidable conundrum. This paper introduce an extension of the division operation from a time-varying perspective. Most problems in science and engineering fields are time-varying, and thus the extension is reasonable and practical. Furthermore, by employing the neat gradient-based neurodynamics (or say, gradient neurodynamics) equation different times, a series of different “symbolic” solutions are proposed. Note that the proposed symbolic solutions have the ability to conquer DBZ problem. The symbolic solutions to DBZ problem may promote the development of more complete singularity-conquering applications.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Kaplan, R.: The Nothing That Is: A Natural History of Zero. Oxford University Press, New York (2000)
Speijer, R.: Defining numbers in terms of their divisors. Nature 461(7260), 37 (2009)
Zhang, Y., Yi, C.: Zhang Neural Networks and Neural-Dynamic Method. Nova Science Publishers, New York (2011)
Zhang, Y., Li, F., Yang, Y., Li, Z.: Different Zhang functions leading to different Zhang-dynamics models illustrated via time-varying reciprocal solving. Appl. Math. Model. 36(9), 4502–4511 (2012)
Zhang, Y., Chen, Z., Chen, K.: Convergence properties analysis of gradient neural network for solving online linear equations. Acta Automatica Sinica 35(8), 1136–1139 (2009)
Zhang, Y., Li, Z., Li, K.: Complex-valued Zhang neural network for online complex-valued time-varying matrix inversion. Appl. Math. Comput. 217(24), 10066–10073 (2011)
Zhang, Y., Li, Z., Guo, D., Li, W., Chen, P.: Z-type and G-type models for time-varying inverse square root (TVISR) solving. Soft Comput. 17(11), 2021–2032 (2013)
Carmesin, H.O.: Multilinear back-propagation convergence theorem. Phys. Lett. A 188(1), 27–31 (1994)
Cai, B., Zhang, Y.: Equivalence of velocity-level and acceleration-level redundancy-resolution of manipulators. Phys. Lett. A 373(38), 3450–3453 (2009)
Hajieghrary, H., Hsieh, M.A., Schwartz, I.B.: Multi-agent search for source localization in a turbulent medium. Phys. Lett. A 380(20), 1698–1705 (2016)
Wang, G., Zhang, W., Lu, J., Zhao, H.: Dispersion and optical gradient force from high-order mode coupling between two hyperbolic metamaterial waveguides. Phys. Lett. A 380(35), 2774–2780 (2016)
Zhang, Y., Zhang, Y., Chen, D., Xiao, Z., Yan, X.: Division by zero, pseudo-division by zero, Zhang dynamics method and Zhang-gradient method about control singularity conquering. Int. J. Syst. Sci. 48(1), 1–12 (2017)
Zhang, Y., Xiao, Z., Guo, D., Mao, M.: Singularity-conquering tracking control of a class of chaotic systems using Zhang-gradient dynamics. IET Control Theor. Appl. 9(6), 871–881 (2015)
Zhang, Y., Yu, X., Yin, Y., Peng, C., Fan, Z.: Singularity-conquering ZG controllers of z2g1 type for tracking control of the IPC system. Int. J. Control 87(9), 1729–1746 (2014)
Zhang, Y., Yu, X., Yin, Y., Xiao, L., Fan, Z.: Using GD to conquer the singularity problem of conventional controller for output tracking of nonlinear system of a class. Phys. Lett. A 377(25–27), 1611–1614 (2013)
Zhang, Y., Qiu, B., Ling, Y., Yang, Z., Peng, C.: What is 1/0 in the general sense of physics, applied computation, and/or electronics? In: The 34th Chinese Control Conference, pp. 1105–1110. IEEE (2015)
Acknowledgments
The work is supported by the National Natural Science Foundation of China (with number 61473323), by the Foundation of Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, China (with number 2013A07), and also by the Laboratory Open Fund of Sun Yat-sen University (with number 20160209). Kindly note that all authors of the paper are jointly of the first authorship, with the following thoughts shared: (1) “By my understanding, science, including mathematics, is tree-like, web-like, city-like, \(\cdots \), growing”, (2) “For a scientifically new thing, there are many levels of contributions: the 1st one is creating; the 2nd one is proving; the 3rd one is applying; \(\cdots \); the nth one is knowing; \(\cdots \)”, (3) “It is better (and may be dangerously better) that a Ph.D. dissertation researches everything about something, absolutely new; it is better that a Master thesis researches some things about something, new; it is at least that a Bachelor thesis researches something about something, relatively new”, (4) “Be a good person for oneself; be a good son, a good husband and a good father (or be a good daughter, a good wife and a good mother) for family; \(\cdots \); be a good labor and a good boss for workunit; \(\cdots \)”, (5) “You are what you have been thinking and doing on the basis of ground”, and (6) “This ends, and that starts”. Thanks a lot and best regards.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Zhang, Y., Gong, H., Li, J., Huang, H., Yin, Z. (2017). Symbolic Solutions to Division by Zero Problem via Gradient Neurodynamics. In: Liu, D., Xie, S., Li, Y., Zhao, D., El-Alfy, ES. (eds) Neural Information Processing. ICONIP 2017. Lecture Notes in Computer Science(), vol 10636. Springer, Cham. https://doi.org/10.1007/978-3-319-70090-8_75
Download citation
DOI: https://doi.org/10.1007/978-3-319-70090-8_75
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-70089-2
Online ISBN: 978-3-319-70090-8
eBook Packages: Computer ScienceComputer Science (R0)