From the fact that the unique solution of a homogeneous linear algebraic system is the trivial one we can obtain the existence of a solution of the nonhomogeneous system. Coefficients of the systems considered are elements of the Colombeau algebra $\overline{ℝ}$ of generalized real numbers. It is worth mentioning that the algebra $\overline{ℝ}$ is not a field.

If $\left\{v_i\right\}$ and $\left\{w_j\right\}$ are two families of unitary bases for ${\mathbf{C}}^n$, and $\theta$ is a fixed number, let $V^n$ and $W^n$ be subspaces of ${\mathbf{C}}^n$ spanned by $[\theta ·n]$ vectors in $\left\{v_i\right\}$ and $\left\{w_j\right\}$ respectively. We study the angle between $V^n$ and $W^n$ as $n$ goes to infinity. We show that when $\left\{v_i\right\}$ and $\left\{w_j\right\}$ arise in certain arithmetically defined families, the angles between $V^n$ and $W^n$ may either tend to $0$ or be bounded away from zero, depending on the behavior of an associated eigenvalue problem.

Let Ω be the spectral unit ball of Mₙ(ℂ), that is, the set of n × n matrices with spectral radius less than 1. We are interested in classifying the automorphisms of Ω. We know that it is enough to consider the normalized automorphisms of Ω, that is, the automorphisms F satisfying F(0) = 0 and F'(0) = I, where I is the identity map on Mₙ(ℂ). The known normalized automorphisms are conjugations. Is every normalized automorphism a conjugation? We show that locally, in a neighborhood of a matrix with...

We investigate the formal matrix ring over $R$ defined by a certain system of factors. We give a method for constructing formal matrix rings from non-negative integer matrices. We also show that the principal factor matrix of a binary system of factors determine the structure of the system.

Let ${\mu }_{n-1}\left(G\right)$ be the algebraic connectivity, and let ${\mu }_1\left(G\right)$ be the Laplacian spectral radius of a $k$-connected graph $G$ with $n$ vertices and $m$ edges. In this paper, we prove that $${\mu }_{n-1}\left(G\right)\ge \frac{2n{k}^2}{(n(n-1)-2m)(n+k-2)+2k^2},$$ with equality if and only if $G$ is the complete graph $K_n$ or $K_n-e$. Moreover, if $G$ is non-regular, then $${\mu }_1\left(G\right)<2\Delta -\frac{2(n\Delta -2m)k^2}{2(n\Delta -2m)(n^2-2n+2k)+n{k}^2},$$ where $\Delta$ stands for the maximum degree of $G$. Remark that in some cases, these two inequalities improve some previously known results.