Composition Series of the Solvable Multiplicative Abelian Groups over a Regular Even nth Roots of Unity: A Classical Approach
Alechenu Benard,
Babayo Muhammed Abdullahi,
Daniel Eneojo Emmanuel
Issue:
Volume 6, Issue 3, June 2021
Pages:
49-54
Received:
27 April 2021
Accepted:
7 June 2021
Published:
29 June 2021
Abstract: Solubility of algebraic structures is what gleaned the introduction of group theory, which later stems the other realms of abstract algebra viz: rings, fields and semigroup theories. The nth roots of unity is found in the most sensitive texts ever in the history of abstract algebra: Cauchy’s, Galois’ and Cayley’s. These three giant group theorists had the common ground of the roots of unity in even the title of their works. The idea is that if the nth roots of unity are solvable by radicals and so do the composition series approach, then all other products of the nth roots of unity - which the unity itself is part of - will automatically be solvable. Hence, all equations that dissolve to the least of the nth roots of unity are solvable by the composition series. This article penciled down how the congruence modulo of arithmetics due to Gauss and Leibnitz were used to break down the nth roots of unity, so that the recursive process can generate the composition series of normal subgroups between the unity and the group itself. Since they are P-Groups, they have normal P-Sylow Subgroups. The normality comes from the Index Theorem. Because they all have index 2 in their P-Groups, they are the maximal proper normal P-Sylow Subgroups and their factor groups are abelian accounting to the solubility of nth roots of unity by composition series. We combine the classical Euler Formula and the De Moivre Theorem to present the solvability of nth roots of unity. The P-Groups over nth roots of unity are multiplicative. nth roots of unity are subsequences of nth roots of unity and it converges to the limit point of the nth roots of unity.
Abstract: Solubility of algebraic structures is what gleaned the introduction of group theory, which later stems the other realms of abstract algebra viz: rings, fields and semigroup theories. The nth roots of unity is found in the most sensitive texts ever in the history of abstract algebra: Cauchy’s, Galois’ and Cayley’s. These three giant group theorists ...
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Mechanical Loading Model of Main Bearing of Wind Turbine Based on Test Bench
Xing Yang,
Tao Zhang,
Lei Li,
Ya-qian Wang
Issue:
Volume 6, Issue 3, June 2021
Pages:
55-65
Received:
21 June 2021
Accepted:
1 July 2021
Published:
7 July 2021
Abstract: Wind-turbine main bearing has to withstand dynamic loads with different directions and different magnitudes in complex environments, and its stable operation has a vital impact on the performance of the entire wind turbine. Therefore, the fatigue strength test of wind turbine bearing is helpful to ensure the normal operation of the whole wind turbine. According to the actual force of main bearing in natural environment, this paper designed a large-scale wind-turbine bearing test bench to detect the deformation performance of the bearing. Through the establishment of the mechanical loading model, the simulation of the actual working conditions of main bearing is realized by the loading of the eight hydraulic cylinders of the test bench, and the radial and axial displacement of the test bearing under different wind conditions are recorded by displacement sensors. A number of temperature sensors are used to monitor the real-time temperature change of the bearing inner ring. The test results show that the loading effect of eight hydraulic cylinders can realize the force of wind turbine bearing under wind load, and the test bench can effectively detect wind-turbine bearings with a diameter of 2.5 m. The mechanical loading method and test results can provide guidance for further inspection of the wind-turbine bearing.
Abstract: Wind-turbine main bearing has to withstand dynamic loads with different directions and different magnitudes in complex environments, and its stable operation has a vital impact on the performance of the entire wind turbine. Therefore, the fatigue strength test of wind turbine bearing is helpful to ensure the normal operation of the whole wind turbi...
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