Archive

Abstract:

This manuscript outlines various work carried out in the field of Distributed Generation (DG). Increase in power consumption and shortage in transmission capabilities are addressed by DGs. In order to maximize the potential benefits, it is imperative to place the DGs at optimal locations and the DGs should have optimal size pertaining to that location. There are several research works that are carried out on the placement and sizing of DGs. Nonetheless, the methodical principle for this issue is still unsettled. Various optimization strategies can be used to obtain the appropriate placement and sizing of distributed generation (DG) in grids. This study provides a comprehensive overview of several DG placement approaches, including stochastic fractal search algorithms, particle swarm optimization, symbiotic search algorithms, opposition-based tuneable chaotic differential evaluation, and more. The benefits and potential uses of each method are briefly covered in this study. The study sheds light on the efforts made to determine the best location and size of DGs.

Keywords:

Distributed Generation,DG Placement Techniques,Optimal Locations,Optimal Size,

Refference:

I. Abou El-Ela, Adel A., Ragab A. El-Sehiemy, and Ahmed Samir Abbas. “Optimal placement and sizing of distributed generation and capacitor banks in distribution systems using water cycle algorithm.” IEEE Systems Journal 12.4 (2018): 3629-3636.
II. Acharya, Naresh, Pukar Mahat, and Nadarajah Mithulananthan. “An analytical approach for DG allocation in primary distribution network.” International Journal of Electrical Power & Energy Systems 28.10 (2006): 669-678.
III. Alam, Afroz, Bushra Zaheer, and Mohammad Zaid. “Optimal placement of DG in distribution system for power loss minimization and voltage profile improvement.” 2018 International Conference on Computing, Power and Communication Technologies (GUCON). IEEE, 2018.
IV. Ali, E. S., S. M. Abd Elazim, and A. Y. Abdelaziz. “Ant Lion Optimization Algorithm for optimal location and sizing of renewable distributed generations.” Renewable Energy 101 (2017): 1311-1324.
V. Aman, M. M., et al. “A new approach for optimum simultaneous multi-DG distributed generation Units placement and sizing based on maximization of system loadability using HPSO (hybrid particle swarm optimization) algorithm.” Energy 66 (2014): 202-215.
VI. Angalaeswari, S., et al. “Hybrid pipso-sqp algorithm for real power loss minimization in radial distribution systems with optimal placement of distributed generation.” Sustainability 12.14 (2020): 5787.
VII. Das, Bikash, V. Mukherjee, and Debapriya Das. “DG placement in radial distribution network by symbiotic organisms search algorithm for real power loss minimization.” Applied Soft Computing 49 (2016): 920-936.
VIII. Devi, Sudha, and M. Geethanjali. “Optimal location and sizing determination of Distributed Generation and DSTATCOM using Particle Swarm Optimization algorithm.” International Journal of Electrical Power & Energy Systems 62 (2014): 562-570.
IX. Devi, S., and M. Geethanjali. “Application of modified bacterial foraging optimization algorithm for optimal placement and sizing of distributed generation.” Expert Systems with Applications 41.6 (2014): 2772-2781.
X. Fandi, Ghaeth, et al. “Voltage regulation and power loss minimization in radial distribution systems via reactive power injection and distributed generation unit placement.” Energies 11.6 (2018): 1399.
XI. Ganthia, B. P., Barik, S., & Nayak, B. (2020). Application of hybrid facts devices in DFIG based wind energy system for LVRT capability enhancements. J. Mech. Cont. Math. Sci, 15(6), 245-256.
XII. Ganthia, B. P., Barik, S. K., & Nayak, B. (2020). Transient Analysis of Grid Integrated Stator Voltage Oriented Controlled Type-Iii DFIG Driven Wind Turbine Energy System. Journal of Mechanics of Continua and Mathematical Sciences, 15(6), 139-157.
XIII. Hung, Duong Quoc, Nadarajah Mithulananthan, and Kwang Y. Lee. “Optimal placement of dispatchable and nondispatchable renewable DG units in distribution networks for minimizing energy loss.” International Journal of Electrical Power & Energy Systems 55 (2014): 179-186.
XIV. Hung, Duong Quoc, Nadarajah Mithulananthan, and R. C. Bansal. “Analytical strategies for renewable distributed generation integration considering energy loss minimization.” Applied Energy 105 (2013): 75-85.
XV. Jordehi, Ahmad Rezaee. “Allocation of distributed generation units in electric power systems: A review.” Renewable and Sustainable Energy Reviews 56 (2016): 893-905.
XVI. Kaushal, Pawan Kumar, and Minal Tomar. “Real and reactive power loss minimization of IEEE-33 bus by optimal DG placement using LSO in RDS.” 2017 International Conference on Energy, Communication, Data Analytics and Soft Computing (ICECDS). IEEE, 2017.
XVII. Kumar, Sajjan, Kamal K. Mandal, and Niladri Chakraborty. “A novel opposition-based tuned-chaotic differential evolution technique for techno-economic analysis by optimal placement of distributed generation.” Engineering Optimization 52.2 (2020): 303-324.
XVIII. Nguyen, Tri Phuoc, and Dieu Ngoc Vo. “A novel stochastic fractal search algorithm for optimal allocation of distributed generators in radial distribution systems.” Applied Soft Computing 70 (2018): 773-796.
XIX. Niknam, Taher, et al. “Optimal operation management of fuel cell/wind/photovoltaic power sources connected to distribution networks.” Journal of Power Sources 196.20 (2011): 8881-8896.
XX. Niknam, T., A. M. Ranjbar, and A. R. Shirani. “Impact of distributed generation on volt/var control in distribution networks.” 2003 IEEE Bologna Power Tech Conference Proceedings,. Vol. 3. IEEE, 2003.
XXI. Prabha, D. Rama, and T. Jayabarathi. “Optimal placement and sizing of multiple distributed generating units in distribution networks by invasive weed optimization algorithm.” Ain Shams Engineering Journal 7.2 (2016): 683-694.
XXII. Prasad, C. Hari, K. Subbaramaiah, and P. Sujatha. “Cost–benefit analysis for optimal DG placement in distribution systems by using elephant herding optimization algorithm.” Renewables: Wind, Water, and Solar 6.1 (2019): 1-12.
XXIII. Prakash, D. B., and C. Lakshminarayana. “Multiple DG placements in radial distribution system for multi objectives using Whale Optimization Algorithm.” Alexandria engineering journal 57.4 (2018): 2797-2806.
XXIV. Raut, Usharani, Sivkumar Mishra, and Debani Prasad Mishra. “An adaptive NSGA II for optimal insertion of distributed generators in radial distribution systems.” 2019 International Conference on Information Technology (ICIT). IEEE, 2019.
XXV. Raut, Usharani, and Sivkumar Mishra. “An improved Elitist–Jaya algorithm for simultaneous network reconfiguration and DG allocation in power distribution systems.” Renewable Energy Focus 30 (2019): 92-106.
XXVI. Reddy, P. Dinakara Prasad, VC Veera Reddy, and T. Gowri Manohar. “Whale optimization algorithm for optimal sizing of renewable resources for loss reduction in distribution systems.” Renewables: wind, water, and solar 4.1 (2017): 1-13.
XXVII. Siahbalaee, Jafar, Neda Rezanejad, and Gevork B. Gharehpetian. “Reconfiguration and DG sizing and placement using improved shuffled frog leaping algorithm.” Electric Power Components and Systems 47.16-17 (2019): 1475-1488.
XXVIII. Teimourzadeh, Hamid, and Behnam Mohammadi-Ivatloo. “A three-dimensional group search optimization approach for simultaneous planning of distributed generation units and distribution network reconfiguration.” Applied Soft Computing 88 (2020): 106012.
XXIX. Tran, Tung The, Khoa Hoang Truong, and Dieu Ngoc Vo. “Stochastic fractal search algorithm for reconfiguration of distribution networks with distributed generations.” Ain Shams Engineering Journal 11.2 (2020): 389-407.
XXX. Truong, Khoa H., et al. “A quasi-oppositional-chaotic symbiotic organisms search algorithm for optimal allocation of DG in radial distribution networks.” Applied Soft Computing 88 (2020): 106067.
XXXI. Yuvaraj, T., K. R. Devabalaji, and K. Ravi. “Optimal allocation of DG in the radial distribution network using bat optimization algorithm.” Advances in power systems and energy management. Springer, Singapore, 2018. 563-569.
XXXII. Zongo, Oscar Andrew, and Anant Oonsivilai. “Optimal placement of distributed generator for power loss minimization and voltage stability improvement.” Energy Procedia 138 (2017): 134-139.

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Abstract:

Estimating the variance matrix has an important role in statistical applications and conclusions, in high–dimensional matrices if the number of variables is greater than the number of observations P > n, the traditional statistical methods are not reliable because they give uncontrolled estimates. Shrinkage methods are used to estimate the high–dimensional variance matrix. In this research, the high–dimensional variance matrix was estimated using the robust Nonparametric method Minimum Regularized Covariance Determinant (MRCD), which is based on Mahalanobis distance, and compared with the variance matrix estimated by the Oracle method, which is based on the Frobenius criterion.

Keywords:

Frobenius,High–Dimensional,Minimum Regularized Covariance Determinant,Mahalanobis,Oracle,Parameter regulation,Shrinkage,

Refference:

I. F. Abdul–Hammeed, and M. Sabah, : ‘Compared with genetic algorithm Fast-MCD-Nested Extension and neural network multilayer Back propagation’. JOURNAL OF ECONOMIC & ADMINISTRATIVE SCIENCE. Jun. No 22(89), 381-395, (2016).
II. F. Virgile, V. Gael, T. Benjamin, and Bertrand Thirion. : ‘Detecting outlying Subjects in High-Dimensional Neuroimaging Datasets with Regularized Minimum Covariance Determinant’. pp. 264-271. https://hal.inria.fr/inria-00626857. 10.1007/978-3-642-23626-6_33
III. I. Clifferd, : ‘High Dimensional Covariance Matrix Estimation’. Department of Statistics, London School. http://stats.lse.ac.uk.
IV. J. Brian Williamson, : ‘Shrinkage Estimators for high-dimensional Covariance matrices’. POMONA COLLEGE , April 4, (2014). 10.1109/ICASSP.2009.4960239
V. K. Jan, and H. Jaroslav, : ‘Robust Regularized Discriminant Analysis Based on Implicit Weighting’. Technical report No.v-1241. December (2016). http://www.nusl.cz/ntk/nusl-262425
VI. K. Jan, T. Jurjen Duintjer, and S. Anna, : ‘Robustness of High-Dimensional Data’.
Mining.Kalina@cs.cas.cz. https://www.semantis/scholarory
VII. L. Olivier, W. Michael, : ‘Shrinkage Estimation of large covariance
matrices: keep it simple’. statistician. university of Zurich . Journal of Multivariate Analysis. 186, (2021) 104796. 10.1016/j.jmva.2021.104796
VIII. M. Abdul – Hammeed,and F. Abdul – Hammeed, : ‘Estimated between the two-stage summation shrinkage for the variance of a normal distribution and for equal sizes of the two samples’. Baghdad science journal. Jun. No 1009, (2011).
IX. M. Hubert, and M. Debruyne, : ‘Minimum Covariance Determinant’. Wiley Interdisciplinary Reviews:Computional Statistics. 2(2010). Pp.- 36-34. https://wis.kuleuven.be/stat/robust/papers/2010/wire-mcd.pdf
X. O. Ledoit ,and M. Wolf , : ‘Quadratic Shrinkage for Large Covariance Matrices’. University of Zurich , November (2019). http://dx.doi.org/10.2139/ssrn.3486378
XI. O. Ledoit ,and M. Wolf. : ‘A well-conditioned estimator for large-dimensional covariance matrices’. Journal of Multivariate Analysis. 88(2) (2004), pp. 365-411. 10.1016/S0047-259X(03)00096-4
XII. R. Maronnan, and R.H. Zamar. : ‘Robust Estimates of Location and Dispersion for High-Dimensional Datasets’. Technometrics. 44(4), 307-317 (2002). https://www.jstor.org/stable/1271538
XIII. P. Rousseeuw , S . Vanduffel and T. Verdonckl. : ‘Minimum Regularized Covariance Determinant Estimatimater’. june 1. (2018). **
XIV. P. Rousseeuw, V. Steven, and V. Tim. : ‘The Minimum Regularized Covariance Determinant Estimator’. ar Xiv:1701.07086v3, November 29 (2018). 10.2139/ssrn.2905259
XV. P. Rousseeuw, and D. Van. : ‘Afast algorithm for the Minimum Covariance Determinant estimator’. Technometrics. 41(3), (1991), pp. 212-223. doi.org/10.2307/1270566
XVI. Won J. H, Lim J. Kim S., J. Rajaratnan. : ‘Condition-number regularized covariance estimation’. J. R. Stat. Ser B (stat.Methodol) 75 (3), (2013) 427-450. doi.org/10.1111/j.1467-9868.2012.01049.x
XVII. Yilun C., Ami wiesel, Alfred O. Hero III. : ‘Shrinkage Estimtion of high Dimensional Covariance Matrices’. International Conference on Acoustics, Speech and Signal Processing. April (2009) 10.1109/ICASSP.2009.4960239
XVIII. Yilun C., Ami Wiesel, Alfredo. : ‘Robust Shrinkage Estimtion of high Dimensional Covariance Matrices’. arXiv:1009.5331v1 [stat.ME]. 27 sep (2010). 10.1109/TSP.2011.2138698
XIX . Zongliang Hu, Kai Dong, Wenlin Dai and Tiejan Tong. : ‘Acomparison of Methods for Estimating the Determinent of High-Dimensional Covariance Matrix’. The International Journal of Biostatistics. September, (2017). doi.org/10.1515/ijb-2017-0013

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Abstract:

This study conducts a bibliometric analysis to explore the evolution and practical applications of legged robots equipped with single-degree-of-freedom mechanisms from 2010 to 2023. Through comprehensive methodologies involving renowned academic databases such as Scopus, the research examines 127 relevant articles, employing statistical analysis and network assessments to discern trends and contributors in the field. Results indicate a peak in publication volume in 2019, with India emerging as the leading contributor, followed by Romania and China. The findings provide valuable insights into the global research landscape of legged robotics, highlighting key advancements and contributors and paving the way for future developments in the field.

Keywords:

Citation,Co-occurrences,Degrees of Freedom,Legged Robots,Walking Mechanisms,

Refference:

I. Armada, M. A., de González Santos, P., Ottaviano, E., Ceccarelli, M., & Tavolieri, C. (2005). Kinematic and dynamic analyses of a pantograph-leg for a biped walking machine. In Climbing and Walking Robots: Proceedings of the 7th International Conference CLAWAR 2004 (pp. 561-568). Springer Berlin Heidelberg.
II. Desai, Shivamanappa G., Anandkumar R. Annigeri, and A. TimmanaGouda. “Analysis of a new single degree-of-freedom eight link leg mechanism for walking machine.” Mechanism and machine theory 140 (2019): 747-764. 10.1016/j.mechmachtheory.2019.06.002
III. Frank, C. Modern Robotics-Mechanics, Planning, and Control. Cambridge University Press, 2017.
IV. Fukuoka, Y., Kimura, H., & Cohen, A. H. (2003). Adaptive dynamic walking of a quadruped robot on irregular terrain based on biological concepts. The International Journal of Robotics Research, 22(3-4), 187-202. 10.1177/0278364903022003004
V. Godoy, J. C., Campos, I. J., Pérez, L. M., & Muñoz, L. R. (2018). Nonanthropomorphic exoskeleton with legs based on eight-bar linkages. International Journal of Advanced Robotic Systems, 15(1), 1729881418755770. 10.1177/1729881418755770

VI. Ishihara, Hidenori, and Kiyoshi Kuroi. “A four-leg locomotion robot for heavy load transportation.” 2006 IEEE/RSJ International Conference on intilligent and robots and systems .IEEE,2006. 10.1109/IROS.2006.282379
VII. Jansen, Theo. The great pretender. 010 Publishers, 2007.
VIII. Jansen, Theo. The great pretender. 010 Publishers, 2007.
IX. Kashem, Saad Bin Abul, et al. “An experimental study of the amphibious robot inspired by biological duck foot.” 2018 IEEE 12th International Conference on Compatibility, Power Electronics and Power Engineering (CPE-POWERENG 2018). IEEE, 2018. 10.1109/CPE.2018.8372507
X. Kashem, Saad Bin Abul, et al. “Design and implementation of a quadruped amphibious robot using duck feet.” Robotics 8.3 (2019): 77. 10.3390/robotics8030077
XI. Kim, H., Lee, D., Jeong, K., & Seo, T. (2015). Water and ground-running robotic platform by repeated motion of six spherical footpads. IEEE/ASME Transactions on Mechatronics, 21(1), 175-183. 10.1109/TMECH.2015.2435017
XII. Kulandaidaasan Sheba, J., Elara, M. R., Martínez-García, E., & Tan-Phuc, L. (2016). Trajectory generation and stability analysis for reconfigurable klann mechanism based walking robot. Robotics, 5(3), 13. 10.3390/robotics5030013
XIII. Liang, C., Ceccarelli, M., Takeda, Y. “Operation Analysis of a Chebyshev-Pantograph Leg Mechanism for a Single DOF Biped Robot.” Frontiers of Mechanical Engineering, vol. 7, no. 4, 2012, pp. 357–370. 10.1007/s11465-012-0340-5
XIV. Lockhande, N. G., and V. B. Emche. “Mechanical spider by using klann mechanisms.” International Journal of Mechanical Engineering and Computer Applications 1.5 (2013): 13-16.
XV. McCarthy, J. M., & Chen, K. Design of Mechanical Walking Robots. MDA, Press, 2021.
XVI. Núñez-Altamirano, Diego A., Felipe J. Torres, and Ignacio Juárez-Campos. “Kinematics of a Reconfigurable Robotic Leg based on the inverse Peaucellier-Lipkin mechanism.” 2019 IEEE International Autumn Meeting on Power, Electronics and Computing (ROPEC). IEEE, 2019. 10.1109/ROPEC48299.2019.9057073
XVII. Patnaik, Lalit, and Loganathan Umanand. “Kinematics and dynamics of Jansen leg mechanism: A bond graph approach.” Simulation Modelling Practice and Theory 60 (2016): 160-169. 10.1016/j.simpat.2015.10.003
XVIII. Rajkumar, A. “A microcontroller based spider bot using Klann mechanism.” AIP Conference Proceedings. Vol. 2460. No. 1. AIP Publishing, 2022. 10.1063/5.0096353
XIX. Regulan, Gopi Krishnan, Ganesan Kaliappan, and M. Santhakumar. “Development of an amphibian legged robot based on Jansen mechanism for exploration tasks.” Advancements in Automation, Robotics and Sensing: First International Conference, ICAARS 2016, Coimbatore, India, June 23-24, 2016, Revised Selected Papers. Springer Singapore, 2016.10.1007/978-981-10-2845-8_7
XX. Shah, Rushil, et al. “Advancement and application of Theo Jansen linkages: A review.” AIP Conference Proceedings. Vol. 2855. No. 1. AIP Publishing, 2023. 10.1063/5.0169581
XXI. Sheba, Jaichandar Kulandaidaasan, et al. “Design and evaluation of reconfigurable Klann mechanism based four legged walking robot.” 2015 10th International Conference on Information, Communications and Signal Processing (ICICS). IEEE, 2015. 10.1109/ICICS.2015.7459939
XXII. Silva, Manuel Fernando, and JA Tenreiro Machado. “A literature review on the optimization of legged robots.” Journal of Vibration and Control 18.12 (2012): 1753-1767.
XXIII. Varma, DS Mohan. “Synthesis and Analysis of Jansen’s Leg-Based Mechanism for Gait Rehabilitation.” Mechanism and Machine Science: Select Proceedings of Asian MMS 2018. Springer Singapore, 2021. 10.1007/978-981-15-4477-4_22

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Abstract:

In this article, we extract the diverse solitary wave solutions to the time-fractional perturbed nonlinear Schrödinger equation describing the dynamics of optical solitons travelling through nonlinear optical fibers. The nonlinear fractional differential equation is transformed into a nonlinear differential equation using a traveling wave transformation relating to the beta derivative. After that, the resulting equation is explained using the extended Riccati equation method. Abundant soliton and soliton-type solutions are extracted, comprising trigonometric and hyperbolic functions. The nature of the solutions varies qualitatively depending on distinct parameters. Additionally, graphical representations of the constructed solutions exhibit various physical forms, including kink, bell-shaped, periodic, anti-coupon etc. Moreover, the achieved solutions play a significant role in interpreting wave propagation studies and are essential for validating numerical and experimental findings in the fields of nonlinear optics, quantum mechanics, engineering, etc.

Keywords:

Beta Derivative,Extended Riccati Equation method,Optical Solitons,Time-fractional Perturbed Nonlinear Schrödinger Equation,Traveling Wave Transformation,

Refference:

I. Akbar, M. A., & Khatun, M. M. (2023). Optical soliton solutions to the space–time fractional perturbed Schrödinger equation in communication engineering. Optical and Quantum Electronics, 55(7), 645. 10.1007/s11082-023-04911-9
II. Ali, A., Ahmad, J., & Javed, S. (2023). Solitary wave solutions for the originating waves that propagate of the fractional Wazwaz-Benjamin-Bona-Mahony system. Alexandria Engineering Journal, 69, 121-133. 10.1016/j.aej.2023.01.063
III. Atangana, A., & Alqahtani, R. T. (2016). Modelling the spread of river blindness disease via the Caputo fractional derivative and the beta-derivative. Entropy, 18(2), 40. 10.3390/e18020040
IV. Atangana, A., Baleanu, D., & Alsaedi, A. (2016). Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal. Open Physics, 14(1), 145-149. 10.1515/phys-2016-0010
V. Beghami, W., Maayah, B., Bushnaq, S., & Abu Arqub, O. (2022). The Laplace optimized decomposition method for solving systems of partial differential equations of fractional order. International Journal of Applied and Computational Mathematics, 8(2), 52. 10.1007/s40819-022-01256-x
VI. Bekir, A., Aksoy, E., & Cevikel, A. C. (2015). Exact solutions of nonlinear time fractional partial differential equations by sub‐equation method. Mathematical Methods in the Applied Sciences, 38(13), 2779-2784. 10.1002/mma.3260
VII. Bekir, A., Guner, O., & Cevikel, A. (2016). The exp-function method for some time-fractional differential equations. IEEE/CAA Journal of Automatica Sinica, 4(2), 315-321. 10.1109/JAS.2016.7510172
VIII. Bilal, M., & Ren, J. (2022). Dynamics of exact solitary wave solutions to the conformable time-space fractional model with reliable analytical approaches. Optical and Quantum Electronics, 54, 1-19. 10.1007/s11082-021-03408-7
IX. Bilal, M., Ren, J., Inc, M., & Alhefthi, R. K. (2023). Optical soliton and other solutions to the nonlinear dynamical system via two efficient analytical mathematical schemes. Optical and Quantum Electronics, 55(11), 938. 10.1007/s11082-023-05103-1
X. Chen, W., Sun, H., & Li, X. (2022). Fractional derivative modeling in mechanics and engineering. Beijing, China: Springer.
XI. Esra Köse, G., Oruç, Ö., & Esen, A. (2022). An application of Chebyshev wavelet method for the nonlinear time fractional Schrödinger equation. Mathematical Methods in the Applied Sciences, 45(11), 6635-6649. 10.1002/mma.8196
XII. Islam, T., Akbar, M. A., & Azad, A. K. (2018). Traveling wave solutions to some nonlinear fractional partial differential equations through the rational (G′/G)-expansion method. Journal of Ocean Engineering and Science, 3(1), 76-81. 10.1016/j.joes.2017.12.003
XIII. Islam, M. T., Akter, M. A., Gómez-Aguilar, J. F., & Akbar, M. A. (2022). Novel and diverse soliton constructions for nonlinear space–time fractional modified Camassa–Holm equation and Schrodinger equation. Optical and Quantum Electronics, 54(4), 227. 10.1007/s11082-022-03602-1

XIV. Khater, M. M., Attia, R. A., & Lu, D. (2018). Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions. Mathematical and Computational Applications, 24(1), 1. 10.3390/mca24010001
XV. Kudryashov, N. A., & Biswas, A. (2022). Optical solitons of nonlinear Schrödinger’s equation with arbitrary dual-power law parameters. Optik, 252, 168497. 10.1016/j.ijleo.2021.168497
XVI. Laskin, N. (2002). Fractional Schrödinger equation. Physical Review E, 66(5), 056108. 10.1103/PhysRevE.66.056108
XVII. Lu, D., Wang, J., Arshad, M., & Ali, A. (2017). Fractional reduced differential transform method for space-time fractional order heat-like and wave-like partial differential equations. Journal of Advanced Physics, 6(4), 598-607. 10.1166/jap.2017.1383
XVIII. Odabasi, M., & Misirli, E. (2018). On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Mathematical Methods in the Applied Sciences, 41(3), 904-911. 10.1002/mma.3533
XIX. Okposo, N. I., Veeresha, P., & Okposo, E. N. (2022). Solutions for time-fractional coupled nonlinear Schrödinger equations arising in optical solitons. Chinese Journal of Physics, 77, 965-984. 10.1016/j.cjph.2021.10.014
XX. Owyed, S., Abdou, M. A., Abdel-Aty, A., & Dutta, H. (2020). Optical solitons solutions for perturbed time fractional nonlinear Schrodinger equation via two strategic algorithms. Aims Math, 5(3), 2057-2070. 10.3934/math.2020136
XXI. Riaz, M. B., Atangana, A., Jahngeer, A., Jarad, F., & Awrejcewicz, J. (2022). New optical solitons of fractional nonlinear Schrodinger equation with the oscillating nonlinear coefficient: A comparative study. Results in Physics, 37, 105471. 10.1016/j.rinp.2022.105471
XXII. Rizvi, S. T. R., Seadawy, A. R., Younis, M., Ahmad, N., & Zaman, S. (2021). Optical dromions for perturbed fractional nonlinear Schrödinger equation with conformable derivatives. Optical and Quantum Electronics, 53(8), 477. 10.1007/s11082-021-03126-0
XXIII. Sarwar, A., Gang, T., Arshad, M., Ahmed, I., & Ahmad, M. O. (2023). Abundant solitary wave solutions for space-time fractional unstable nonlinear Schrödinger equations and their applications. Ain Shams Engineering Journal, 14(2), 101839. 10.1016/j.asej.2022.101839
XXIV. Valentim, C. A., Rabi, J. A., & David, S. A. (2021). Fractional mathematical oncology: On the potential of non-integer order calculus applied to interdisciplinary models. Biosystems, 204, 104377. 10.1016/j.biosystems.2021.104377
XXV. Wang, F., Salama, S. A., & Khater, M. M. (2022). Optical wave solutions of perturbed time-fractional nonlinear Schrödinger equation. Journal of Ocean Engineering and Science. 10.1016/j.joes.2022.03.014
XXVI. Wazwaz, A. M. (2022). Bright and dark optical solitons of the (2+ 1)-dimensional perturbed nonlinear Schrödinger equation in nonlinear optical fibers. Optik, 251, 168334. 10.1016/j.ijleo.2021.168334
XXVII. Younis, M., ur Rehman, H., Rizvi, S. T. R., & Mahmood, S. A. (2017). Dark and singular optical solitons perturbation with fractional temporal evolution. Superlattices and Microstructures, 104, 525-531. 10.1016/j.spmi.2017.03.006
XXVIII. Zaman, U. H. M., Arefin, M. A., Akbar, M. A., & Uddin, M. H. (2023). Utilizing the extended tanh-function technique to scrutinize fractional order nonlinear partial differential equations. Partial Differential Equations in Applied Mathematics, 8, 100563. 10.1016/j.padiff.2023.100563
XXIX. Zhu, S. D. (2008). The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2+1)-dimensional Boiti-Leon-Pempinelle equation. Chaos, Solitons & Fractals, 37(5), 1335-1342. 10.1016/j.chaos.2006.10.015

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Abstract:

Over the last several years, there has been a significant increase in the amount of focus placed on the infrastructure development of smart cities. The primary issue that academics are attempting to address is the issue of energy efficiency. One of these issues was the identification of anomalies in energy usage, which was an essential component that needed to be taken into consideration when managing energy-saving systems that were efficient, hence lowering the total energy consumption and carbon emissions. Therefore, the proposal of a strong approach that is based on the Internet of Things (IoT) might provide more relevance for the identification of abnormal consumption in buildings and the provision of this information to customers and governments so that it can be handled in an appropriate manner to minimize payments. Consequently, the purpose of this work is to explore three different optimization methods, namely ADAM, AadMax, and Nadam, and to advocate for an optimization approach that makes use of the LSTM algorithm to identify anomalies. Statistical modelling techniques such as ARIMA and SARIMAX are used for the purpose of time series forecasting. The findings of the anomaly detection system reveal that the best results are obtained by using LSTM in conjunction with Nadar. The MSE and RMSE values reached were 0.15348 and 0.02356 respectively. Additionally, the ARIMA model yields the best overall results, with the AIC value being 0.13859 and the MSE value being 300.94365 correspondingly. Confirmation of the suggested model's dependability and flexibility in optimizing anomaly detection is provided by this particular fact.

Keywords:

Anomaly Detection System,Abnormal Consumption,Energy-saving Systems,Statistical Modelling Techniques,Time Series Forecasting.,

Refference:

I. Abdulwahid, M. M., Al-Ani, O. A. S., Mosleh, M. F., & Abd-Alhameed, R. A. : ‘Investigation of millimeter-wave indoor propagation at different frequencies’. In 2019 4th Scientific International Conference Najaf (SICN). (2019, April). (pp. 25-30). IEEE.‏
II. Abdulwahid, M. M., Al-Ani, O. A. S., Mosleh, M. F., & Abd-Alhameed, R. A. : ‘A Comparison between Different C-band and mmWave band Frequencies for Indoor Communication’. J. Commun., 14(10), (2019). pp. 892-899.‏
III. Abdulwahid, M. M., Al-Hakeem, M. S., Mosleh, M. F., & Abd-alhmeed, R. A. : ‘Investigation and optimization method for wireless AP deployment based indoor network’. In IOP Conference Series: Materials Science and Engineering. Vol. 745, No. 1. (2020, February). pp. 012031. IOP Publishing.‏
IV. Abdulwahid, M. M., & Kurnaz, S. : ‘The channel WDM system incorporates of Optical Wireless Communication (OWC) hybrid MDM-PDM for higher capacity (LEO-GEO) inter-satellite link’. Optik. VoL. 273 (2023). 170449.‏ doi.org/10.1016/j.ijleo.2022.170449
V. Abd-Alhameed, R. A., Abdulwahid, M. M., & Mosleh, M. F. : ‘Effects of Antenna Directivity and Polarization on Indoor Multipath Propagation Characteristics for different mmWave frequencies’.‏ Informatica 2(1). 2021. pp. 20-28.
VI. Ali, O. M. A., Kareem, S. W., & Mohammed, A. S. : ‘Evaluation of Electrocardiogram Signals Classification Using CNN, SVM, and LSTM Algorithm: A review’. In 2022 8th International Engineering Conference on Sustainable Technology and Development (IEC), pp. 185-191. IEEE, 2022.‏

VII. Almetwali, A. S., Bayat, O., Abdulwahid, M. M., & Mohamadwasel, N. B. : ‘Design and Analysis of 50 Channel by 40 Gbps DWDM-RoF System for 5G Communication Based on Fronthaul Scenario’. In Proceedings of Third Doctoral Symposium on Computational Intelligence. (2023). (pp. 109-122). Springer, Singapore.‏
VIII. Alhamadani, N. B., & Abdelwahid, M. M. : ‘Implementation of microstrip patch antenna using MATLAB. Informatica’. Journal of Applied Machines Electrical Electronics Computer Science and Communication Systems. 2(1). (2021). pp. 29-35.‏
IX. Alsalemi A, Himeur Y, Bensaali F, et al. : ‘Achieving domestic energy efficiency using micro‐moments and intelligent recommendations. IEEE Access. Vol.8. 2020. pp.15047‐15055,
X. Ampountolas, A. : ‘Modeling and Forecasting Daily Hotel Demand: A Comparison Based on SARIMAX, Neural Networks, and GARCH Models’. Forecasting, Vol.3, No. 3, pp.580-595., 2021
XI. A. Mosavi, A. Bahmani. : ‘Energy consumption prediction using machine learning: A review’. Preprints 2019, doi.org/10.20944/preprints201903.0131.v1, 2019.
XII. Alsalemi A, Himeur Y, Bensaali F, et al. : ‘Achieving domestic energy efficiency using micro‐moments and intelligent recommendations. IEEE Access. Vol.8. 2020. pp.15047‐15055.
XIII. Box, G. E., and Jenkins, G. M. : ‘Time series analysis: forecasting and control’. Holden dsy. Inc. California, 1976.
XIV. Burhan, I. M., Al-Hakeem, M. S., Abdulwahid, M. M., & Mosleh, M. F. : ‘Investigating the Access Point height for an indoor IOT services’. In IOP Conference Series: Materials Science and Engineering. Vol. 881, No. 1, (2020, July). pp. 012116. IOP Publishing.‏
XV. Buzau, M. M., Tejedor-Aguilera, J., et al. : ‘Hybrid deep neural networks for detection of non-technical losses in electricity smart meters. IEEE Transactions on Power Systems, Vol. 35, No. 2, 2019.‏ pp.1254-1263.
XVI. Chou, J. S., & Telaga, A. S. : ‘Real-time detection of anomalous power consumption’. Renewable and Sustainable Energy Reviews, Vol. 33. 2014. pp.400-411.,
XVII. Farsi, M., Hosahalli, D., Manjunatha, B. R., Gad, I., et al. : ‘Parallel genetic algorithms for optimizing the SARIMA model for better forecasting of the NCDC weather data. Alexandria Engineering Journal, Vol.60, No. 1, (2021).‏ pp. 1299-1316.

XVIII. Feng, L., Xu, S., Zhang, L., Wu, J., Zhang, J., et al. : ‘Anomaly detection for electricity consumption in cloud computing: framework, methods, applications, and challenges’. EURASIP Journal on Wireless Communications and Networking, Vol. 1. (2020). pp.1-12, ‏
XIX. F. Abayaje, S. A. Hashem, H. S. Obaid, Y. S. Mezaal, and S. K. Khaleel. : ‘A miniaturization of the UWB monopole antenna for wireless baseband transmission’. Periodicals of Engineering and Natural Sciences, vol. 8. no. 1. (2020). pp. 256-262.
XX. G. Zhao, L. Xing. ‘Reliability analysis of IoT systems with competitions from cascading probabilistic function dependence. Reliab. Eng. Syst. Saf., Vol. 198. (2020). pp.106812. https://doi.org/10.1016/j.ress.2020.106812
XXI. Himeur Y, Elsalemi A, Bensaali F, Amira A. : ‘Robust event‐based non‐intrusive appliance recognition using multi‐scale wavelet packet tree and ensemble bagging tree. Appl Energy. Val. 267. (2020). pp.114877.
XXII. Himeur, Y., Alsalemi, A., Bensaali, et al. : ‘A novel approach for detecting anomalous energy consumption based on micro-moments and deep neural networks’. Cognitive Computation, Vol. 12. No. 6. (2020). pp. 1381-1401.
XXIII. Himeur, Y., Alsalemi, A., Bensaali, F., at al. : ‘Smart power consumption abnormality detection in buildings using micromoments and improved K‐nearest neighbors. International Journal of Intelligent Systems, Vol. 36, No. 6. (2012). pp. 2865-2894.
XXIV. Hyndman, R. J., and Athanasopoulos, G. : ‘Forecasting: principles and practice. OTexts. 2018.‏
XXV. H. A. Hussein, Y. S. Mezaal, and B. M. Alameri. : ‘Miniaturized microstrip diplexer based on FR4 substrate for wireless communications’. Elektron. Ir Elektrotech. Vol. 27 No. 5. (2021) . doi.org/10.5755/j02.eie.28942
XXVI. I. O. Essiet, Y. Sun, Z. Wang. ‘Optimized energy consumption model for smart home using improved differential evolution algorithm’. Energy Vol. 172. (2019). pp.354–365. https://doi.org/10.1016/j.
XXVII. Jamal, S. A., Ibrahim, A. A., Abdulwahid, M. M., & Wasel, N. B. M. (2020). ‘Design and implementation of multilevel security-based home management system’.‏ International Journal of Advanced Trends in Computer Science and Engineering. Vol. 9(4). (2020). pp. 5716-5720. doi.org/10.30534/ijatcse/2020/224942020
XXVIII. J. Ali and Y. Miz’el. ‘A new miniature Peano fractal-based bandpass filter design with 2nd harmonic suppression 3rd IEEE International Symposium on Microwave.’ Antenna, Propagation and EMC Technologies for Wireless Communications, Beijing, China, 2009.
XXIX. Lin, G., & Claridge, D. E. : ‘A temperature-based approach to detect abnormal building energy consumption’. Energy and Buildings, Vol. 93. (2015). pp. 110-118.
XXX. Liu, J., Shahroudy, A., Xu, D., et al. : ‘Spatio-temporal lstm with trust gates for 3d human action recognition’. In European conference on computer vision. (2016). pp. 816-833. Springer, Cham.,
XXXI. Liu Y, Geng G, Gao S, Xu W. : Non‐intrusive energy use monitoring for a group of electrical appliances’. IEEE Trans Smart Grid, Vol.9, No. 4. (2018). pp. 3801‐3810,
XXXII. Luo, Y., Li, W., & Qiu, S. : ‘Anomaly detection based latency-aware energy consumption optimization for IoT data-flow services’. Sensors, Vol.20. No. 1. (2019). pp.122.
XXXIII. Malki, A., Atlam, E. S., and Gad, I. : ‘Machine learning approach of detecting anomalies and forecasting time-series of IoT devices’. Alexandria Engineering Journal, Vol. 61 No. 11. (2022). pp. 8973-8986.
XXXIV. Mohsen, D. E., Abbas, E. M., & Abdulwahid, M. M. : ‘Design and Implementation of DWDM-FSO system for Tbps data rates with different atmospheric Attenuation’. In 2022 International Congress on Human-Computer Interaction, Optimization and Robotic Applications (HORA) (2022, June). (pp. 1-7). IEEE.‏ 10.1109/HORA55278.2022.9799974
XXXV. Mohsen, D. E., Abbas, E. M., & Abdulwahid, M. M. : ‘Performance Analysis of OWC System based (S-2-S) Connection with Different Modulation Encoding’. International Journal of Intelligent Systems and Applications in Engineering. 11(4s). (2023). pp. 400-408.
XXXVI. Monner, D., and Reggia, J. A. : ‘A generalized LSTM-like training algorithm for second-order recurrent neural networks. Neural Networks, Vol. 25. (2012). pp.70-83.
XXXVII. Mustapha, A., Mohamed, L., and Ali, K. : ‘Comparative study of optimization techniques in deep learning: Application in the ophthalmology field’. In Journal of Physics: Conference Series, Vol. 1743, No. 1. (2021) pp. 012002. IOP Publishing.
XXXVIII. M. Kh. : ‘Cybercrime Challenges in Iraqi Academia: Creating Digital Awareness for Preventing Cybercrimes.’ International Journal of Cyber Criminology. vol. 16, no. 2. (2022). pp. 1–14.
XXXIX. M. Q. Mohammed. : ‘HARNESSING CLOUD OF THING AND FOG COMPUTING IN IRAQ: ADMINISTRATIVE INFORMATICS SUSTAINABILITY’. Journal of Mechanics of Continua and Mathematical Sciences. vol. 19, no. 2. (2024) pp. 66–78. 10.26782/jmcms.2024.02.00004
XL. M. S. Jameel, Y. S. Mezaal, and D. C. Atilla. : ‘Miniaturized coplanar waveguide-fed UWB Antenna for wireless applications’. Symmetry. vol. 15, no. 3. (2023). pp. 633. doi.org/10.3390/sym15030633
XLI. M. S. Shareef et al. : ‘Cloud of Things and fog computing in Iraq: Potential applications and sustainability.’ : Heritage and Sustainable Development. vol. 5, no. 2. (2023). pp. 339–350.
XLII. M. Zekic´-Susˇac, S. Mitrovic´, A. Has. : ‘Machine learning based system for managing energy efficiency of public sector as an approach towards smart cities’. Int. J. Inf. Manage. Vol 58. (2021) doi.org/10.1016/j.ijinfomgt.2020.102074
XLIII. Sardianos C, Varlamis I, Chronis C, et al. : ‘A model for predicting room occupancy based on motion sensor data’. In: 2020 IEEE International Conference on Informatics, IoT, and Enabling Technologies (ICIoT). (2020). pp. 394‐399.
XLIV. Sardianos C, Varlamis I, Dimitrakopoulos G, et al. : ‘REHAB‐C: recommendations for energy HABits change’. Future Gener Comput Syst. Vol. 112. (2020). pp. 394‐407. 10.1016/j.future.2020.05.041
XLV. Sohani, A., Sayyaadi, H., Cornaro, C., Shahverdian, et. al. : ‘Using machine learning in photovoltaics to create smarter and cleaner energy generation systems: A comprehensive review’. Journal of Cleaner Production. Vol 364 (2022). doi.org/10.1016/j.jclepro.2022.132701
XLVI. S. A. Abdulameer. : ‘Security Readiness in Iraq: Role of the Human Rights Activists’. International Journal of Cyber Criminology. vol. 16. no. 2. (2022) pp. 1–14.
XLVII. S. I. Yahya et al.. ‘A New Design Method for Class-E Power Amplifiers Using Artificial Intelligence Modeling for Wireless Power Transfer Applications’. Electronics. vol. 11, no. 21 (2022). pp. 3608.
XLVIII. S. Roshani et al. : ‘Design of a compact quad-channel microstrip diplexer for L and S band applications’. Micromachines (Basel), vol. 14, no. 3. (2023). doi.org/10.3390/mi14030553
XLIX. T. Abd, Y. S. Mezaal, M. S. Shareef, S. K. Khaleel, H. H. Madhi, and S. F. Abdulkareem. : ‘Iraqi e-government and cloud computing development based on unified citizen identification’. Periodicals of Engineering and Natural Sciences. vol. 7. no. 4. (2019) pp. 1776–1793.
L. Vagropoulos, S. I., Chouliaras, G. I., Kardakos, E. G., et al. : ‘Comparison of SARIMAX, SARIMA, modified SARIMA and ANN-based models for short-term PV generation forecasting’. In 2016 IEEE International Energy Conference (ENERGYCON), pp. 1-6. IEEE. 10.1109/ENERGYCON.2016.7514029
LI. V. Marinakis, H. Doukas. : ‘An advanced IoT-based system for intelligent energy management in buildings’. Sensors, Vol.18, No. 2, (2018). pp.610. https://doi.org/10.3390/s18020610
LII. Wu, D., Zhang, L., and Lin, L. : ‘Based on the moving average and target motion information for detection of weak small target’. In 2018 International Conference on Intelligent Transportation, Big Data & Smart City (ICITBS), pp. 641-644. IEEE, 2018.‏
LIII. Yang, L., & Yang, H. : ‘A Combined ARIMA-PPR Model for Short-Term Load Forecasting’. In 2019 IEEE Innovative Smart Grid Technologies-Asia (ISGT Asia). IEEE. (2019). pp. 3363-3367.
LIV. Y. S. Mezaal and H. T. Eyyuboglu. : ‘A new narrow band dual-mode microstrip slotted patch bandpass filter design based on fractal geometry’. In 2012 7th International Conference on Computing and Convergence Technology (ICCCT). IEEE. (2012). pp. 1180-1184.
LV. Y. S. Mezaal, and J. K. Ali. : ‘A new design of dual band microstrip bandpass filter based on Peano fractal geometry: Design and simulation results’. Presented at the 2013 13th Mediterranean Microwave Symposium (MMS), IEEE. 2013. 10.1109/MMS.2013.6663140
LVI. Y. S. Mezaal and S. F. Abdulkareem. : ‘New microstrip antenna based on quasi-fractal geometry for recent wireless systems’. In 2018 26th Signal Processing and Communications Applications Conference (SIU). IEEE. (2018) pp. 1-4.
LVII. Y. S. Mezaal et al. : ‘Cloud computing investigation for cloud computer networks using cloudanalyst’. Journal of Theoretical and Applied Information Technology, vol. 96. no. 20. (2018)
LVIII. Y.S. Mezaal et al. : ‘Cloud computing investigation for cloud computer networks using cloudanalyst’. Journal of Theoretical and Applied Information Technology. vol. 96, no. 20, (2018)
LIX. Y. S. Mezaal, H. H. Saleh, and H. Al-Saedi. ‘New compact microstrip filters based on quasi fractal resonator’. Advanced Electromagnetics. vol. 7. no. 4. (2018) pp. 93-102.
LX. Y. S.Mezaal, Eyyuboglu, H. T., &Ali, J. K. Wide bandpass and narrow bandstop microstrip filters based onHilbert fractal geometry: design and simulation results. PloS one, 9(12),e115412, 2014.
LXI. Zaal, R. M., Mustafa, F. M., Abbas, E. I., Mosleh, M. F., & Abdulwahid, M. M. : ‘Real measurement of optimal access point localizations’. In IOP Conference Series: Materials Science and Engineering. Vol. 881. No. 1. (2020). pp. 012119). IOP Publishing.

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Abstract:

In this paper, the author stated and proved Bhattacharyya’s Theorem: √{-(x² + y² )} = -√(x² + y²). With the help of this theorem, the author finds the square root of any negative number introducing Fermat’s last theorem without using the concept of complex numbers. The author has introduced Fermat’s Last Theorem in Bhattacharyya’s Theorem to find the square root of any negative number in real numbers in a very simple way. Indeed it is a new invention in mathematics in this era.

Keywords:

Extended form of Pythagoras Theorem,Fermat’s Last Theorem,Pythagoras Theorem,Rectangular Bhattacharyya’s Co-ordinate System,Theory of Dynamics of Numbers,

Refference:

I. A. Harripersaud. : ‘The quadratic equation concept’. American Journal of Mathematics and Statistics. 11.(3). pp. 67-71. (2021).

II. B. B. Datta, & A. N. Singh. : ‘History of Hindu Mathematics, A source book’. Mumbai, Maharashtra: Asia Publishing House. (1938).

III. G. H. Hardy and E. M. Wright. : ‘An Introduction to the Theory of Numbers’. Sixth Edition. Page 245 – 247.

IV. H. Lee Price, Frank R. Bernhart. : ‘Pythagorean Triples and a New Pythagorean Theorem’. arXiv:math/0701554 [math.HO]. 10.48550/arXiv.math/0701554
V. L. Nurul , H., D. : ‘Five New Ways to Prove a Pythagorean Theorem’. International Journal of Advanced Engineering Research and Science. Volume 4, issue 7. (2017). pp.132-137. 10.22161/ijaers.4.7.21

VI. Makbule Gözde DİDİŞ KABAR. : ‘A Thematic Review of Quadratic Equation Studies in The Field of Mathematics Education’. Participatory Educational Research. (PER)Vol.10(4). (2023). pp. 29-48. 10.17275/per.23.58.10.4

VII. Manjeet Singh. : ‘Transformation of number system’. International Journal of Advance Research, Ideas and Innovations in Technology. Volume 6, Issue 2. (2020). pp 402-406. 10.13140/RG.2.2.33484.77442

VIII. M. Sandoval-Hernandez, H. Vazquez-Leal, U. Filobello-Nino , Elisa De-Leo-Baquero, Alexis C. Bielma-Perez, J.C. Vichi-Mendoza , O. Alvarez-Gasca, A.D. Contreras-Hernandez, N. Bagatella-Flores , B.E. Palma-Grayeb, J. Sanchez-Orea, L. Cuellar-Hernandez. : ‘The Quadratic Equation and its Numerical Roots’. IJERT. Volume 10, Issue 06 (June 2021), 301-305. 10.17577/IJERTV10IS060100

IX. Prabir Chandra Bhattacharyya, : ‘AN INTRODUCTION TO THEORY OF DYNAMICS OF NUMBERS: A NEW CONCET’. J. Mech. Cont. & Math. Sci., Vol.-17, No.-1, pp 37-53, January (2022). 10.26782/jmcms.2022.01.00003

X. Prabir Chandra Bhattacharyya, : ‘A NOVEL CONCEPT OF THE THEORY OF DYNAMICS OF NUMBERS AND ITS APPLICATION IN THE QUADRATIC EQUATION’. J. Mech. Cont. & Math. Sci., Vol.-19, No.-2, pp 93-115, February (2024). 10.26782/jmcms.2024.02.00006

XI. Prabir Chandra Bhattacharyya, : ‘A NEW CONCEPT TO PROVE, √−1 = −1 IN BOTH GEOMETRIC AND ALGEBRAIC METHODS WITHOUT USING THE CONCEPT OF IMAGINARY NUMBERS’. J. Mech. Cont. & Math. Sci., Vol.-18, No.-9, pp 20-43. 10.26782/jmcms.2023.09.00003

XII. Prabir Chandra Bhattacharyya, : ‘AN INTRODUCTION TO RECTANGULAR BHATTACHARYYA’S CO-ORDINATES: A NEW CONCEPT’. J. Mech. Cont. & Math. Sci., Vol.-16, No.-11, pp 76. November (2021). 10.26782/jmcms.2021.11.00008
XIII. Prabir Chandra Bhattacharyya, : ‘A NEW CONCEPT OF THE EXTENDED FORM OF PYTHAGORAS THEOREM’. J. Mech. Cont. & Math. Sci., Vol.-18, No.-04, April (2023) pp 46-56. 10.26782/jmcms.2023.04.00004.

XIV. S. Mahmud. : ‘Calculating the area of the Trapezium by Using the Length of the Non Parallel Sides : ‘A New Formula for Calculating the area of Trapezium’. International Journal of Scientific and Innovative Mathematical Research. volume 7, issue 4, pp. 25-27. (2019) 10.20431/2347- 3142.0704004

XV. T. A. Sarasvati Amma, : ‘Geometry in Ancient and Medieval India’. pp. – 219. Motilal Banarasidass Publishers Pvt. Ltd. Delhi.

XVI. William Robert.: ‘An Overview of Number System’. RRJSMS. Volume 8. Issue 4. April, 2022. 10.4172/ J Stats Math Sci.8.4.002.

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