Volume 5 , Issue 2 , PP: 69-80, 2023 | Cite this article as | XML | Html | PDF | Full Length Article
Prem Kumar Singh 1 *
Doi: https://doi.org/10.54216/JNFS.050207
Recently, a problem is addressed while dealing the data with Non-Euclidean Geometry and its characterization. The mathematician found negation of fifth postulates of Euclidean geometry easily and called as Non-Euclidean geometry. However Riemannian provided negation of second postulates also which still considered as Non-Euclidean. In this case the problem arises what will happen in case negation of other Euclid Postulates exists. Same time total total or partial negation of Euclid postulates fails as hybrid Geometry. It become more crucial in case the data is unknown, incomplete or exists beyond the three-way space as heteroclinic pattern. To understand this problem, the current paper tried to distinguish Euclidean, Non-Euclidean, Anti-Geometry, Neutrogeometry and Turiyam or Unknown geometry using the complement operator with an example.
Consciousness , Euclidean geometry , knowledge representation , NeutroGeometry , Non-Euclidean , Turiyam Geometry , Unknown Graph. ,
[1] Birkhoff G.D., “A Set of Postulates for Plane Geometry (Based on Scale and Protractors)”. Annals of Mathematics, Vol. 33, 1932.
[2] Russell B., “Introduction: An essay on the foundations of geometry”. Cambridge University Press, 1897.
[3] Coxeter H.S.M., “Non-Euclidean Geometry. University of Toronto Press, 1942. reissued 1998 by Mathematical Association of America.
[4] Singh PK, “Data with Non-Euclidean Geometry and its Characterization”. Journal of Artificial Intelligence and Technology, Jan 2022, Volume 2, Issue 1, pp. 3-8, doi : 10.37965/jait.2021.12001
[5] Lobachevsky N., “Pangeometry, Translator and Editor: A. Papadopoulos. Heritage of European Mathematics Series". European Mathematical Society, Vol. 4, 2010.
[6] Jürgen J., “Riemannian Geometry and Geometric Analysis”. Berlin: Springer-Verlag (2002)
[7] Bhattacharya S., “A model to a Smarandache Geometry”. 2004, http://fs.unm.edu/ModelToSmarandacheGeometry.pdf
[8] Popov, M. R., “The Smarandache Non-Geometry. Abstracts of Papers Presented to the American Mathematical Society Meetings, Vol. 17, Issue 3, pp. 595, 1996.
[9] Kuciuk L., and Antholy M., “An introduction to the Smarandache geometries. JP Journal of Geometry & Topology”, Vol. 5, Issue 1, 77-81, 2005, http://fs.unm.edu/IntrodSmGeom.pdf
[10] Smarandache F., “NeutroGeometry & AntiGeometry are alternatives and generalizations of the NonEuclidean Geometries”. Neutrosophic Sets and Systems, Vol. 46, pp. 456-476, 2021. http://fs.unm.edu/NSS/NeutroGeometryAntiGeometry31.pdf
[11] Singh P. K., “AntiGeometry and NeutroGeometry Characterization of Non-Euclidean Data Sets”. Journal of Neutrosophic and Fuzzy Systems, Vo 1, Issue 1, pp. 24-33, 2021, doi: https://doi.org/10.54216/JNFS.0101012
[12] Singh P. K., “NeutroAlgebra and NeutroGeometry for Dealing Heteroclinic Patterns.” In: Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebras, IGI Global Publishers, 2021, Chapter 6, doi: 10.4018/978-1-6684-3495-6
[13] Granados C., “A note on AntiGeometry and NeutroGeometry and their application to real life.”Neutrosophic Sets and Systems, Vol. 49, pp. 579-593, 2022. doi: 10.5281/zenodo.6466520
[14] Deng X., and Papadimitriou C. H. (1990) Exploring an unknown graph," Journal of graph Theory, 32 (3):265-297,
[15] Singh P.K., “Turiyam set a fourth dimension data representation. Journal of Applied Mathematics and Physics, Vol. 9, Issue 7, pp. 1821-1828, 2021, DOI: 10.4236/jamp.2021.97116
[16] Singh P.K., “Fourth dimension data representation and its analysis using Turiyam Context”. Journal of Computer and Communications, Vol. 9, Issue 6, pp. 222-229, 2021 doi: 10.4236/jcc.2021.96014
[17] Singh P. K., “ Data with Turiyam Set for Fourth Dimension Quantum Information Processing”. Journal of Neutrosophic and Fuzzy Systems, Vol 1, Issue 1, pp. 9-23, 2021. doi: https://doi.org/10.54216/JNFS.010101
[18] Singh PK, Ahmad KD, Bal M, and Aswad M., “On The Symbolic Turiyam Rings.” Journal of Neutrosophic and Fuzzy Systems, Vol. 1 , No. 2 , pp. 80-88, 2021, Doi :
https://doi.org/10.54216/JNFS.010204
[19] Bal M., Singh P. K., and Ahmad K. D., “A Short Introduction To The Symbolic Turiyam Vector Spaces and Complex Numbers.” Journal of Neutrosophic and Fuzzy Systems, Vol. 2, Issue 1, pp. 76-87, 2022. doi : https://doi.org/10.54216/JNFS.020107
[20] Bal M., Singh P. K., and Ahmad K. D., “A Short Introduction To The Concept Of Symbolic Turiyam Matrix. Journal of Neutrosophic and Fuzzy Systems, Vol. 2, Issue 1, pp. 88-99, 2022 doi : https://doi.org/10.54216/JNFS.020108
[21] Bal M., Singh P. K., and Ahmad K. D, “An Introduction To The Symbolic Turiyam R-Modules and Turiyam Modulo Integers.” Journal of Neutrosophic and Fuzzy Systems, Vol. 2, Issue 2, pp. 8-19, doi: https://doi.org/10.54216/JNFS.020201
[22] Belnap J. N. , “A useful four-valued logic”. In J. Michael Dunn and George Epstein, editors, Proceedings of the Fifth International Symposium on Multiple-Valued Logic, Modern Uses of Multiple-Valued Logic. Indiana University, D. Reidel Publishing Company, pp 8 -37, 1975.
[23] Font J. M., “Belnap's Four-Valued Logic and De Morgan Lattices.” In: Logic Journal of the IGPL, Vol. 5, no. 3, pp. 1-29, 1997, doi: 10.1093/jigpal/5.3.1-e.
[24] Irmak E., and Kurtuldu, Ö., “Concept of 4th Dimension for Databases.” In: Proceedings of 2015 IEEE 14th International Conference on Machine Learning and Applications (ICMLA), pp. 1159-1162, doi: 10.1109/ICMLA.2015.186.
[25] James A. W., “Hyperbolic Geometry”. Second edition 2005, Springer.
[26] Pandey L. K., Ojha K. K., Singh P.K., Singh C. S., Dwivedi S., Bergey E.A, “Diatoms image database of India (DIDI): a research tool”. Environmental Technology & Innovation, Vol. 5, pp. 148160, 2016. https://doi.org/10.1016/j.eti.2017.02.005
[27] Tixeira da Silva J.A., Vuong Q.H., “The right to refuse unwanted citations: rethinking the culture of science around the citation”. Scientometrics, Vol. 126, pp. 5355–5360, 2021.
[28] Singh P. K., “t-index: Entropy based random document and citation analysis using average h-index.” Scientrometrics, Vol 127, pp. 637-660, 2022, doi: 10.1007/s11192-021-04222-4
[29] Singh, P. K., “Quaternion set for dealing fluctuation in quantum turiyam cognition”. Journal of Neutrosophic and Fuzzy Systems, Vol. 4, Issue 02, pp. 57–64, 2022.https://doi.org/10.54216/JNFS.010101
[30] Singh P. K., “Air Quality Index Analysis Using Single-Valued Neutrosophic Plithogenic Graph for MultiDecision Process.” International Journal of Neutrosophic Sciences, V ol 16, Issue 1, pp. 28-141, 2021, doi:https://doi.org/10.54216/IJNS.160103
[31] Singh PK, “Non-Euclidean, Anti-Geometry and NeutroGeometry Characterization”. International Journal of Neutrosophic Sciences, Vol 18, Issue 3, pp. 8-19, 2022, doi : https://doi.org/10.54216/IJNS.180301
[32] Singh, P. K. (2022) “Four-Way Turiyam set-based human quantum cognition analysis”. Journal of Artificial Intelligence and Technology, Vol. 2, Issue 4, pp. 144-151
[33] Ani A., Mashadi M., Sri G. "Invers Moore-Penrose pada Matriks Turiyam Simbolik Real." Jambura Journal of Mathematics 5, no. 1, pp. 95-114, 2023.
[34] Smarandache, F., “Introduction to the symbolic plithogenic algebraic structures (revisited)”. Neutrosophic Set and System, 53, 653–665, 2023.
[35] Singh Pritpal, “Ambiguous Set Theory: A New Approach to Deal with Unconsciousness and Ambiguousness of Human Perception”. Journal of Neutrosophic and Fuzzy Systems, Vol. 5 , No. 1 , pp. 52-58, 2023
[36] Silva, Manuel. "The Latest Advances on AI and Robotics Applications." Journal of Artificial Intelligence and Technology 2, no. 4, pp. 131-131, 2022.
