Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems

Malkhaz Mumladze

doi:doi:10.11648/j.ijmfs.20200601.12

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Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems

Malkhaz Mumladze

Published in International Journal of Management and Fuzzy Systems (Volume 6, Issue 1)

Received: 26 April 2020 Accepted: 1 June 2020 Published: 15 June 2020

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Abstract

In this paper, we introduced the concept of pseudo-convex open covering of topological spaces and entropy for topological spaces with such covering. Entropy was previously defined only for open coverings of compact topological spaces. It is shown by examples that the classes of topological spaces for which the concept of entropy is defined is quite wide. A discrete random process describing the evolution of the phase space of closed dynamical systems is built. Two random processes are constructed, one in which the elements of the transition matrices depend on the first indices, and the second Markov`s process, one in which the elements of the transition matrices do not depend from the first indices. The construction of transition matrices is based on the fact that the probability of a change in the phase space of a system to another space is proportional to the entropy of this other space. Based on the concept of entropy of topological spaces and on the well-known construction of an infinite product of probability measures which is also probabilistic introduced the concept entropy of trajectory of evolution of phase space of system. Previously, the entropy of the trajectory was defined only for the motion of a structureless material point, in this article is defined entropy of trajectory of evolution of structured objects represented in the form of topological spaces. On the basis of the concept entropy of trajectory, a method is determined for finding the most probable trajectory of evolution of the phase space of a closed system.

Published in	International Journal of Management and Fuzzy Systems (Volume 6, Issue 1)
DOI	10.11648/j.ijmfs.20200601.12
Page(s)	8-13
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Topological Space, Covering, Entropy, Random Process

References

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[2]	J. L Doob Stochastic processes New- York -John Wiley& Sons, London- Chapman& Hall 1953.
[3]	J. Bell Infinite product measures. Departments of Mathematics, university of Toronto, May 10, 2015.
[4]	H. H. Schaefer topological vector spaces, Springer 2012.
[5]	Edwin H. Spanier Algebraic Topology, MeGRAWHILL BOOK COMPANY 1966.
[6]	V. I. Bogachev, Measure Theory Springer 2007.
[7]	R. Ash Probability and measure theory, Academic Press; 2 edition 1999.
[8]	Ya. G. Sinai, “On the Notion of Entropy of a Dynamical System,” Doklady of Russian Academy of Sciences, 1959.
[9]	T. Downarowicz Entropy in dynamical Systems, New Mathematical Monograph, Cambridge University Press, Cambridge 2011.
[10]	Dou Dou, Wen Huang, Kyewon Koh Park Entropy dimension of measure preserving systems cornell university arXiv: 1312.7225v2 [math. DS] 3 Jan 2014.
[11]	Sadahiro SaekA Proof of the Existence of Infinite Product Probability Measures, The American Mathematical Monthly, Vol. 103, No. 8, 1996.
[12]	P. BILLINGSLEY Probability and Measure, Third Edition The Universityof Chicago, A Wiley-Interscience Publication WILEY & JOHN SONS New York, Chichester, Brisbane, Toronto, Singapore 1995.
[13]	E. B. Dynkin and A. A. Yushkevich Markov Processes, English ed. Plenum Press, New York, 1969.
[14]	Ryszard Engelking General topology Heldermann Verlag, 1989.
[15]	Lei ZHANG Leijun LIU Wen LI Sparse Trajectory Prediction Method Based on Entropy Estimation IEICE TRANSACTIONS on Information and Systems Vol. E99-D No. 6. 2016.

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Malkhaz Mumladze. (2020). Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems. International Journal of Management and Fuzzy Systems, 6(1), 8-13. https://doi.org/10.11648/j.ijmfs.20200601.12

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ACS Style

Malkhaz Mumladze. Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems. Int. J. Manag. Fuzzy Syst. 2020, 6(1), 8-13. doi: 10.11648/j.ijmfs.20200601.12

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AMA Style

Malkhaz Mumladze. Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems. Int J Manag Fuzzy Syst. 2020;6(1):8-13. doi: 10.11648/j.ijmfs.20200601.12

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@article{10.11648/j.ijmfs.20200601.12,
  author = {Malkhaz Mumladze},
  title = {Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {6},
  number = {1},
  pages = {8-13},
  doi = {10.11648/j.ijmfs.20200601.12},
  url = {https://doi.org/10.11648/j.ijmfs.20200601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20200601.12},
  abstract = {In this paper, we introduced the concept of pseudo-convex open covering of topological spaces and entropy for topological spaces with such covering. Entropy was previously defined only for open coverings of compact topological spaces. It is shown by examples that the classes of topological spaces for which the concept of entropy is defined is quite wide. A discrete random process describing the evolution of the phase space of closed dynamical systems is built. Two random processes are constructed, one in which the elements of the transition matrices depend on the first indices, and the second Markov`s process, one in which the elements of the transition matrices do not depend from the first indices. The construction of transition matrices is based on the fact that the probability of a change in the phase space of a system to another space is proportional to the entropy of this other space. Based on the concept of entropy of topological spaces and on the well-known construction of an infinite product of probability measures which is also probabilistic introduced the concept entropy of trajectory of evolution of phase space of system. Previously, the entropy of the trajectory was defined only for the motion of a structureless material point, in this article is defined entropy of trajectory of evolution of structured objects represented in the form of topological spaces. On the basis of the concept entropy of trajectory, a method is determined for finding the most probable trajectory of evolution of the phase space of a closed system.},
 year = {2020}
}

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TY  - JOUR
T1  - Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems
AU  - Malkhaz Mumladze
Y1  - 2020/06/15
PY  - 2020
N1  - https://doi.org/10.11648/j.ijmfs.20200601.12
DO  - 10.11648/j.ijmfs.20200601.12
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
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PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20200601.12
AB  - In this paper, we introduced the concept of pseudo-convex open covering of topological spaces and entropy for topological spaces with such covering. Entropy was previously defined only for open coverings of compact topological spaces. It is shown by examples that the classes of topological spaces for which the concept of entropy is defined is quite wide. A discrete random process describing the evolution of the phase space of closed dynamical systems is built. Two random processes are constructed, one in which the elements of the transition matrices depend on the first indices, and the second Markov`s process, one in which the elements of the transition matrices do not depend from the first indices. The construction of transition matrices is based on the fact that the probability of a change in the phase space of a system to another space is proportional to the entropy of this other space. Based on the concept of entropy of topological spaces and on the well-known construction of an infinite product of probability measures which is also probabilistic introduced the concept entropy of trajectory of evolution of phase space of system. Previously, the entropy of the trajectory was defined only for the motion of a structureless material point, in this article is defined entropy of trajectory of evolution of structured objects represented in the form of topological spaces. On the basis of the concept entropy of trajectory, a method is determined for finding the most probable trajectory of evolution of the phase space of a closed system.
VL  - 6
IS  - 1
ER  -

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Author Information

Malkhaz Mumladze

Department of Education, Exact and Natural Sciences, Gori University, Gori, Georgia

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Plain Text BibTeX RIS

APA Style

Malkhaz Mumladze. (2020). Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems. International Journal of Management and Fuzzy Systems, 6(1), 8-13. https://doi.org/10.11648/j.ijmfs.20200601.12

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ACS Style

Malkhaz Mumladze. Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems. Int. J. Manag. Fuzzy Syst. 2020, 6(1), 8-13. doi: 10.11648/j.ijmfs.20200601.12

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AMA Style

Malkhaz Mumladze. Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems. Int J Manag Fuzzy Syst. 2020;6(1):8-13. doi: 10.11648/j.ijmfs.20200601.12

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@article{10.11648/j.ijmfs.20200601.12,
  author = {Malkhaz Mumladze},
  title = {Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems},
  journal = {International Journal of Management and Fuzzy Systems},
  volume = {6},
  number = {1},
  pages = {8-13},
  doi = {10.11648/j.ijmfs.20200601.12},
  url = {https://doi.org/10.11648/j.ijmfs.20200601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijmfs.20200601.12},
  abstract = {In this paper, we introduced the concept of pseudo-convex open covering of topological spaces and entropy for topological spaces with such covering. Entropy was previously defined only for open coverings of compact topological spaces. It is shown by examples that the classes of topological spaces for which the concept of entropy is defined is quite wide. A discrete random process describing the evolution of the phase space of closed dynamical systems is built. Two random processes are constructed, one in which the elements of the transition matrices depend on the first indices, and the second Markov`s process, one in which the elements of the transition matrices do not depend from the first indices. The construction of transition matrices is based on the fact that the probability of a change in the phase space of a system to another space is proportional to the entropy of this other space. Based on the concept of entropy of topological spaces and on the well-known construction of an infinite product of probability measures which is also probabilistic introduced the concept entropy of trajectory of evolution of phase space of system. Previously, the entropy of the trajectory was defined only for the motion of a structureless material point, in this article is defined entropy of trajectory of evolution of structured objects represented in the form of topological spaces. On the basis of the concept entropy of trajectory, a method is determined for finding the most probable trajectory of evolution of the phase space of a closed system.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Entropy of Topological Space S and Evolution of Phase Space of Dynzmical Systems
AU  - Malkhaz Mumladze
Y1  - 2020/06/15
PY  - 2020
N1  - https://doi.org/10.11648/j.ijmfs.20200601.12
DO  - 10.11648/j.ijmfs.20200601.12
T2  - International Journal of Management and Fuzzy Systems
JF  - International Journal of Management and Fuzzy Systems
JO  - International Journal of Management and Fuzzy Systems
SP  - 8
EP  - 13
PB  - Science Publishing Group
SN  - 2575-4947
UR  - https://doi.org/10.11648/j.ijmfs.20200601.12
AB  - In this paper, we introduced the concept of pseudo-convex open covering of topological spaces and entropy for topological spaces with such covering. Entropy was previously defined only for open coverings of compact topological spaces. It is shown by examples that the classes of topological spaces for which the concept of entropy is defined is quite wide. A discrete random process describing the evolution of the phase space of closed dynamical systems is built. Two random processes are constructed, one in which the elements of the transition matrices depend on the first indices, and the second Markov`s process, one in which the elements of the transition matrices do not depend from the first indices. The construction of transition matrices is based on the fact that the probability of a change in the phase space of a system to another space is proportional to the entropy of this other space. Based on the concept of entropy of topological spaces and on the well-known construction of an infinite product of probability measures which is also probabilistic introduced the concept entropy of trajectory of evolution of phase space of system. Previously, the entropy of the trajectory was defined only for the motion of a structureless material point, in this article is defined entropy of trajectory of evolution of structured objects represented in the form of topological spaces. On the basis of the concept entropy of trajectory, a method is determined for finding the most probable trajectory of evolution of the phase space of a closed system.
VL  - 6
IS  - 1
ER  -

Copy | Download