COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

Authors

  • Bernard Alechenu Alechenu Oglekwu Ogbogwu
  • Babayo Muhammed Abdullahi
  • Daniel Eneojo Emmanuel

DOI:

https://doi.org/10.33003/fjs-2021-0502-458

Keywords:

Cauchy Sequence, Rouche Theorem, Class Equation, Aleph Naught

Abstract

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through  the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.

 

 

References

Boyer, H.E. and Cary P.R., (1988).Quenching and control of distortion. Park, OH: ASM International, Materials, 24-25.

Buczek A., and Telejko T. (2013). Investigation of heat transfer coefficient during quenching in various cooling agents, International Journal of Heat and Fluid Flow, 44 (2013) 358–364.

Dodo R.M., Ause T., Dauda E. T., Shehu U. and Popoola A. P. I., (2019). Multi-response optimization of transesterification parameters of mahogany seed oil using grey relational analysis in Taguchi method for quenching application, 5 (8), August 2019, e02167, https://doi.org/10.1016/j.heliyon.2019.e02167

Durowoju, M.O., Adebiyi, K.A., Adekunle, A.S. (2013). Quench Severity of Bioquenchants on medium carbon steel for industrial heat treatment, Annals of Faculty Engineering Hunedoara – International Journal of Engineering, Tome XI (4), 53-58.

Goryushin, V. V., Istomin, N. N., Ksenofontov, A.G., Marsel, A.V. & Yu Shevchenko, S., (1991). Metal Science and Heat Treatment, 41(1-2), 3-7.

Hasan, H. S. Peet, M. J. Jalil, J. M. & Bhadeshia H. K. D. H. (n.d). Heat Transfer Coefficients during Quenching of Steels, available at https://www.phase-trans.msm.cam.ac.uk /2011/transfer_Hasan_2011.pdf , Accessed on 23 Aug, 2018.

Ilo, S. Just, C. & Xhiku F. (2012). Optimisation of multiple quality characteristics of hardfacing using grey-based Taguchi method, Materials and Design, 33, 459–468.

Jayaraman P. and Kumar, L. M. (2014). Multi-response Optimization of Machining Parameters of Turning AA6063 T6 Aluminium Alloy using Grey Relational Analysis in Taguchi Method, Procedia Engineering 97, 197 – 204.

Knothe, G. (2001). Analytical methods used in the production and fuel quality assessment of biodiesel. Trans ASAE 44(2): 193–200.

Komatsu, D., Elki C.S., Lauralice F.C.C., and Geoge, E.T. (2009). “Effect of Corrosion Inhibitors on Cooling Curve Behavior of Soybean Oilâ€Based Quenchantsâ€, Conf. Proceed. New Challenges in Heat Treating and Surface Engineering – Conference in Honor of Božidar LiÅ¡cić, June 9â€12, 2009, Cavtat Croatia, Publ. by Croatian Society for Heat Treatment and Surface Engineering, Zagreb, Croatia, 37â€44.

Liscic, B., Tensi H.M., Canale L.C.F, Totten G.E. (2010). Quenching theory and technology. Boca Raton, FL: CRC Press, 47-54.

MacKenzie, S. (2013). The Mechanism of Quench Oil Oxidation, Houghton International, Inc. Valley Forge, PA, USA 19482.

Pan, L.K. Wang, C. C. Wei, S. L. Sher H. F. (2007). Optimizing multiple quality characteristics via Taguchi method-based Grey analysis, Journal of Materials Processing Technology 182, 107–116.

Rajan, R. R. N. and Rajesh R. (2017). Multi-objective optimisation of Mahua oil transesterification using response surface methodology and grey relational analysis, Int. J. of Enterprise Network Management, 8, (4), 340 – 360.

Ramesh, G. and Prabhu, K. N. (2014). Wetting and Cooling Performance of Vegetable Oils during Quench Hardening, Heat Transfer—Asian Research, 00 (00): 1-16.

Siatis, N.G., Kimbaris, A.C., Pappas, C.S., Tarantilis, P.A. and Polissiou, M.G. (2006). Improvement of biodiesel production based on the application of ultrasound: Monitoring of the procedure by FTIR spectroscopy. Journal of the American Oil Chemists’ Society, 83(1) 34-43.

Simêncio, E., Otero, R. Canale, L. and Totten, G. (2016). Stabilization of vegetable oil-based quenchants to thermal-oxidative degradation: experimental strategy and effect of oxidation on quenching performance. La Metallurgia Italiana, 3.

Sudeepan, J. Kumarb, K. Barmanc, T. K. and Sahoo, P. (2014). Study of tribological behavior of ABS/ CaCO3 composite using grey relational analysis, Procedia Materials Science, 6, 682 – 691.

Published

2021-07-17

How to Cite

Alechenu , B., Abdullahi, B. M., & Emmanuel, D. E. (2021). COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS. FUDMA JOURNAL OF SCIENCES, 5(2), 462 - 469. https://doi.org/10.33003/fjs-2021-0502-458