Examples of associative algebras for which the T-space of central polynomials is not finitely based
In 1988 (see [7]), S. V. Okhitin proved that for any field k of characteristic zero, the T-space CP(M2(k)) is finitely based, and he raised the question as to whether CP(A) is finitely based for every (unitary) associative algebra A for which 0 ≠ T(A) ⊊ CP(A). V. V. Shchigolev (see [9], 2001) showed that for any field of characteristic zero, every T-space of k0〈X〉 is finitely based, and it follows from this that every T-space of k1〈X〉 is also finitely based. This more than answers Okhitin’s question (in the affirmative) for fields of characteristic zero.For a field of characteristic 2, the infinite-dimensional Grassmann algebras, unitary and nonunitary, are commutative and thus the T-space of central polynomials of each is finitely based.We shall show in the following that if p > 2 and k is an arbitrary field of characteristic p, then neither CP(G0) nor CP(G) is finitely based, thus providing a negative answer to Okhitin’s question.
The finite basis property of T-spaces over fields of characteristic zero
In this paper we prove that any T-space over a field of characteristic zero has a finite basis. This result generalizes Kemer's theorem on the existence of a finite basis for any system of associative identities over a field of characteristic zero.
Thermal-mechanical fully coupled analysis of high-speed angular contact ball bearings
The thermal and mechanical properties of angular contact ball bearings are critical to their operating accuracy and service life. On the basis of the theory of dynamics and frictional heat generation, this work establishes a thermal-mechanical fully coupled model. The high-speed motion characteristics, temperature, and dynamic stiffness are analyzed, and the simulation results are compared with the theoretical calculation and experimental results. Then, the temperature and axial deformation of the bearings under different rotation speeds are analyzed, and the laws of mechanical and thermal characteristics of the bearings are obtained. Results show that an increase in the rotation speed of the inner ring causes the temperature of each component to increase, with the contact area of the balls showing the greatest increase in temperature. An increase in rotation speed also increases the axial deformation of the inner ring.