Robust of Interval Dynamic Systems

The frequency and algebraic directions of researches of robust stability are considered. Frequency or Tsypkin-Polyaka’s direction is considered briefly in a survey order. The algebraic or Kharitonov’s direction is considered more more widely, namely basic provisions and results of the Algebraic method of robust stability of interval dynamic systems developed within development algebraic or Kharitonov’s direction of robust stability are presented. Fundamental works of V. L. Kharitonov since issue have caused a huge flow of the publications connected with extreme relevance of the solution of problems of a robustness of systems. So far from a circle of problems of a robustness many issues of robust stability are resolved. Discrete analogs and versions of theorems of Kharitonov are received. Frequency conditions of robust stability are considered and solved in Ya. Z. Tsypkin, B. T. Polyak, Yu. I. Neymark works. However in a problem of robust stability not all issues are so far resolved, especially big contradictions have arisen in a continuous case. Also the tasks considered here for interval matrixes and polyhedrons of matrixes haven’t been solved. In work the theorem like the third theorem of Kharitonov which cancels counterexamples to former known results in this direction is formulated and proved and also on its basis the costal theorem for polyhedrons of matrixes is proved. The new costal theorem also cancels counterexamples for this case. To the main theorem of the considered algebraic method the specifying remark is formulated that in the absence of a full set of four angular polynoms of Kharitonov of a condition of this theorem are necessary, but can be insufficient for stability of system. Determination of angular separate coefficients of a characteristic polynom of system is generally carried out by means of use of methods of nonlinear programming. For a discrete case the discrete analog of the theorem of Kharitonov which is received on the basis of Schur’s theorem is presented. At the same time, the concepts of points and intervals of a variabless used for the theorem of an analog of a continuous case are entered. The algorithm of definition of a robustness of discrete systems is formulated. The main results, the Algebraic method of robust stability developed by the author are illustrated with cancellation of the counterexamples widely known from scientific literature.

Schur's theorem and its relation to the closure properties of the non-abelian tensor product

We show that the Schur multiplier of a Noetherian group need not be finitely generated. We prove that the non-abelian tensor product of a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group is a polycyclic (resp. polycyclic-by-finite) group. We also prove new versions of Schur's theorem.

AN APPROACH TO CAPABLE GROUPS AND SCHUR’S THEOREM

Podoski and Szegedy [‘On finite groups whose derived subgroup has bounded rank’, Israel J. Math.178 (2010), 51–60] proved that for a finite group $G$ with rank $r$, the inequality $[G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$ holds. In this paper we omit the finiteness condition on $G$ and show that groups with finite derived subgroup satisfy the same inequality. We also construct an $n$-capable group which is not $(n+1)$-capable for every $n\in \mathbf{N}$.