We continue the study started recently by Agore, Bontea and Militaru in ``Classifying bicrossed products of Hopf algebras'' (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler's Hopf algebras. Equivalently, we classify all bicrossed products $H_4 \bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4, H_4, \triangleright , \triangleleft )$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for any $\lambda \in k$, we describe by generators and relations the associated bicrossed product, $\mathcal {H}_{16, \lambda }$. This is a $16$-dimensional, pointed, unimodular and non-semisimple Hopf algebra. A Hopf algebra $E$ factorizes through $H_4$ and $H_4$ if and only if $ E \cong H_4 \otimes H_4$ or $E \cong {\mathcal H}_{16, \lambda }$. In the last step we classify these quantum groups by proving that there are only three isomorphism classes represented by: $H_4 \otimes H_4$, ${\mathcal H}_{16, 0}$ and ${\mathcal H}_{16, 1} \cong D(H_4)$, the Drinfel'd double of $H_4$. The automorphism group of these objects is also computed: in particular, we prove that ${\rm Aut}_{\rm Hopf}( D(H_4))$ is isomorphic to a semidirect product of groups, $k^{\times } \rtimes \mathbb {Z}_2$.
The flexible supports make it possible to decrease the forces transmitted between the working machines and their foundations. As rotating machinery, the hydrodynmic bearings exhibiting compliance and considerable damping together with high load capacity are ofen used for this purpose. Nevertheless, on certain operating conditions, the hydraulic forces produced in the thin oil film can destabilize the rotor equilibrion position, which consequently leads to development of the rotor self-excited vibration of large amplitude. There are several ways how to increase the critical speed of the rotor rotation at which the self-excited vibration starts. Recently, the attention was focused on increasing damping in the rotor supports or on modification of the bearings geometry. At present time there are available many active control devices working on various physical principles. In this paper the research of the concept based on controlled kinematic excitation of the bearings shells is carried out. The performed computer simulations prove that the investigated technique enables to reduce amplitude of the self-excited vibration and to increase the rotor angular speed of rotation, at which the self-excited vibration begins. Suppression of the rotor oscillation is always connected with increasing the force by which the turning rotor acts on its stationary part. The developed computational procedures and results of the performed analyses are intended for finding properties and behaviour of the active control devices for mitigation of excessive vibration of rotating machines and to contribute to their proper design in this way. and Obsahuje seznam literatury
In the real-life engineering practice, non-linear regression models have to be designed rather often. To ensure their technical or physical feasibility, such models may, in addition, require another coupling condition. This paper describes two procedures for designing a specific non-linear model using AI methods. A Radial Basis Functions (RBF) based optimization is presented of the model using Genetic Algorithms (GA). The problem solved was based on practical measurements and experiments. The results presented in the paper can be applied to many technical problems in mechanical and civil engineering and other engineering fields. and Obsahuje seznam literatury