Abstract In paper [1] , the following property of a square matrix A is claimed: if the matrix D A... more Abstract In paper [1] , the following property of a square matrix A is claimed: if the matrix D A is a Q -matrix for every positive diagonal matrix D then A 2 is a P 0 + -matrix. We show by an explicit example that this claim is wrong even for the case of 2 × 2 matrices.
In this paper, we provide new stability conditions for second-order dynamical systems. We study t... more In this paper, we provide new stability conditions for second-order dynamical systems. We study the stability conditions for a parameter-dependent second order model and the minimal decay rate for some variations of positive parameters. A certain way of gyroscopic stabilization of an unstable parameter-dependent system is also considered. Mathematics Subject Classification (2010) 15A18 · 15A12 · 34D10
We prove that for any circulant matrix C of size n× n with the monic characteristic polynomial p(... more We prove that for any circulant matrix C of size n× n with the monic characteristic polynomial p(z), the spectrum of its (n− 1)× (n− 1) submatrix Cn−1 constructed with first n− 1 rows and columns of C consists of all critical points of p(z). Using this fact we provide a simple proof for the Schoenberg conjecture recently proved by R.Pereira and S.Malamud. We also prove full generalization of a higher order Schoenberg-type conjecture proposed by M. de Bruin and A. Sharma and recently proved by W.S.Cheung and T.W.Ng. in its original form, i.e. for polynomials whose mass centre of roots equals zero. In this particular case, our inequality is stronger than it was conjectured by de Bruin and Sharma. Some Schmeisser’s-like results on majorization of critical point of polynomials are also obtained.
We provide a counterexample to some statements dealing with a sufficient property for the square ... more We provide a counterexample to some statements dealing with a sufficient property for the square of a matrix to be a $P_0^+$ -matrix.
A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named J--si... more A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named J--sign-symmetric. The spectrum of a J--sign-symmetric irreducible matrix is studied under assumptions that its second compound matrix is also J--sign-symmetric and irreducible. The conditions, when such matrices have complex eigenvalues on the largest spectral circle, are given. The existence of two positive simple eigenvalues $\lambda_1 > \lambda_2 > 0$ of a J--sign-symmetric irreducible matrix A is proved under some additional conditions. The question, when the approximation of a J--sign-symmetric matrix with a J--sign-symmetric second compound matrix by strictly J--sign-symmetric matrices with strictly J--sign-symmetric compound matrices is possible, is also studied in this paper.
Abstract In paper [1] , the following property of a square matrix A is claimed: if the matrix D A... more Abstract In paper [1] , the following property of a square matrix A is claimed: if the matrix D A is a Q -matrix for every positive diagonal matrix D then A 2 is a P 0 + -matrix. We show by an explicit example that this claim is wrong even for the case of 2 × 2 matrices.
In this paper, we provide new stability conditions for second-order dynamical systems. We study t... more In this paper, we provide new stability conditions for second-order dynamical systems. We study the stability conditions for a parameter-dependent second order model and the minimal decay rate for some variations of positive parameters. A certain way of gyroscopic stabilization of an unstable parameter-dependent system is also considered. Mathematics Subject Classification (2010) 15A18 · 15A12 · 34D10
We prove that for any circulant matrix C of size n× n with the monic characteristic polynomial p(... more We prove that for any circulant matrix C of size n× n with the monic characteristic polynomial p(z), the spectrum of its (n− 1)× (n− 1) submatrix Cn−1 constructed with first n− 1 rows and columns of C consists of all critical points of p(z). Using this fact we provide a simple proof for the Schoenberg conjecture recently proved by R.Pereira and S.Malamud. We also prove full generalization of a higher order Schoenberg-type conjecture proposed by M. de Bruin and A. Sharma and recently proved by W.S.Cheung and T.W.Ng. in its original form, i.e. for polynomials whose mass centre of roots equals zero. In this particular case, our inequality is stronger than it was conjectured by de Bruin and Sharma. Some Schmeisser’s-like results on majorization of critical point of polynomials are also obtained.
We provide a counterexample to some statements dealing with a sufficient property for the square ... more We provide a counterexample to some statements dealing with a sufficient property for the square of a matrix to be a $P_0^+$ -matrix.
A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named J--si... more A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named J--sign-symmetric. The spectrum of a J--sign-symmetric irreducible matrix is studied under assumptions that its second compound matrix is also J--sign-symmetric and irreducible. The conditions, when such matrices have complex eigenvalues on the largest spectral circle, are given. The existence of two positive simple eigenvalues $\lambda_1 > \lambda_2 > 0$ of a J--sign-symmetric irreducible matrix A is proved under some additional conditions. The question, when the approximation of a J--sign-symmetric matrix with a J--sign-symmetric second compound matrix by strictly J--sign-symmetric matrices with strictly J--sign-symmetric compound matrices is possible, is also studied in this paper.
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Papers by Volha Kushel