Left semi-braces and solutions of the Yang–Baxter equation

Skew braces and the Yang-Baxter equation

Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation. To study non-involutive solutions one needs skew braces, a non-commutative analog of braces. In this talk we discuss basic properties of skew braces and how these structures are related to the Yang-Baxter equation. We also discuss interesting connections between skew braces and...

On the semi-dynamical reflection equation: solutions and structure matrices

Explicit solutions of the non-constant semi-dynamical reflection equation are constructed, together with suitable parametrizations of their structure matrices. Considering the semi-dynamical reflection equation with rational non-constant Arutyunov-Chekhov-Frolov structure matrices, and a specific meromorphic ansatz, it is found that only two sets of the previously found constant solutions are e...

Left invertibility and invertible solutions of the Lyapunov equation

If the solution of a Lyapunov or Riccati equation corresponding to a finite-dimensional system is invertible, then this leads to additional properties of this system. For infinite-dimensional systems these results do not need to hold. We show that it is necessary to put some additional conditions on the spectrum of the system operator A. For instance, if this operator has compact resolvent, the...

Diagonal and Monomial Solutions of the Matrix Equation AXB=C

In this article, we consider the matrix equation $AXB=C$, where A, B, C are given matrices and give new necessary and sufficient conditions for the existence of the diagonal solutions and monomial solutions to this equation. We also present a general form of such solutions. Moreover, we consider the least squares problem $min_X |C-AXB |_F$ where $X$ is a diagonal or monomial matrix. The explici...

WAVELET SOLUTIONS OF THE KLEIN-GORDON EQUATION