The finite-step realizability of the joint spectral radius of a pair of d×d matrices one of which being rank-one

We study the finite-step realizability of the joint/generalized spectral radius of a pair of real square matrices ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$, one of which has rank 1, where $2?d<+\infty$. Let $\rho (A)$ denote the spectral radius of a square matrix A. Then we prove that there always exists a finite-length word $(i_1^1\text{,}\dots \text{,}{i}_m^1)\in \{1\text{,}2{\}}^m$, for some finite $m?1$, such that$\sqrt[m]{\rho \left({\mathrm{S}}_{i_1^1}?{\mathrm{S}}_{i_m^1}\right)}={\sup }_{n?1}\left\{{\displaystyle \underset{(i_1\text{,}\dots \text{,}{i}_n)\in \{1\text{,}2{\}}^n}{\max }}\sqrt[n]{\rho ({\mathrm{S}}_{i_1}?{\mathrm{S}}_{i_n})}\right\}\text{.}$In other words, there holds the spectral finiteness property for $\{{\mathrm{S}}_1\text{,}{\mathrm{S}}_2\}$. Explicit formula for computation of the joint spectral radius is derived. This implies that the stability of the switched system induced by $\{{\mathrm{S}}_1\text{,}{\mathrm{S}}_2\}$ is algorithmically decidable in this case.     

The inverse of nonsymmetric two-level Toeplitz operator matrices

The Gohberg–Semencul formula allows one to express the entries of the inverse of a Toeplitz matrix using only a few entries (the first row and the first column) of the inverse matrix, under some nonsingularity condition. In this paper we will provide a two variable generalization of the Gohberg–Semencul formula in the case of a nonsymmetric two-level Toeplitz matrix with a symbol of the form $f(z_1\text{,}{z}_2)=\frac{1}{\overline{P(z_1\text{,}{z}_2)}Q(z_1\text{,}{z}_2)}$ where $P(z_1\text{,}{z}_2)$ and $Q(z_1\text{,}{z}_2)$ are stable polynomials of two variables. We also consider the case of operator valued two-level Toeplitz matrices. In addition, we propose an equation solver involving two-level Toeplitz matrices. Numerical results are included.     

Group classification,optimal system and optimal reductions of a class of Klein Gordon equations

Complete symmetry analysis is presented for non-linear Klein Gordon equations $u_{\mathit{tt}}=u_{\mathit{xx}}+f(u)$. A group classification is carried out by finding $f(u)$ that give larger symmetry algebra. One-dimensional optimal system is determined for symmetry algebras obtained through group classification. The subalgebras in one-dimensional optimal system and their conjugacy classes in the corresponding normalizers are employed to obtain, up to conjugacy, all reductions of equation by two-dimensional subalgebras. This is a new idea which improves the computational complexity involved in finding all possible reductions of a PDE of the form $F(x\text{,}t\text{,}u\text{,}{u}_x\text{,}{u}_t\text{,}{u}_{\mathit{xx}}\text{,}{u}_{\mathit{tt}}\text{,}{u}_{\mathit{xt}})=0$ to a first order ODE. Some exact solutions are also found.     

Matrix completion problems over integral domains: The case with a diagonal of prescribed blocks

Let $\mathfrak{R}$ be an arbitrary integral domain, let $\wedge =\{{\lambda }_1\text{,}\dots \text{,}{\lambda }_n\}$ be a multiset of elements of $\mathfrak{R}$, let $\sigma$ be a permutation of $\{1\text{,}\dots \text{,}k\}$ let $n_1\text{,}\dots \text{,}{n}_k$ be positive integers such that $n_1+?+n_k=n$, and for $r=1\text{,}\dots \text{,}k$ let $A_r\in {\mathfrak{R}}^{n_r\times n_{\sigma (r)}}$. We are interested in the problem of finding a block matrix $Q={\left[Q_{\mathit{rs}}\right]}_{r\text{,}s=1}^k\in {\mathfrak{R}}^{n\times n}$ with spectrum $\mathrm{\Lambda }$ and such that $Q_{r\sigma (r)}=A_r$ for $r=1\text{,}\dots \text{,}k$. Cravo and Silva completely characterized the existence of such a matrix when $\mathfrak{R}$ is a field. In this work we construct a solution matrix $Q$ that solves the problem when $\mathfrak{R}$ is an integral domain with two exceptions: (i) $k=2$; (ii) $k\ge 3$, $\sigma (r)=r$ and $n_r>n/2$ for some $r$.What makes this work quite unique in this area is that we consider the problem over the more general algebraic structure of integral domains, which includes the important case of integers. Furthermore, we provide an explicit and easy to implement finite step algorithm that constructs an specific solution matrix (we point out that Cravo and Silva’s proof is not constructive).