We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n×n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel’s theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert...

The point equation of the associated curve of the indefinite numerical range is derived, following Fiedler’s approach for definite inner product spaces. The classification of the associated curve is presented in the $3\times 3$ indefinite case, using Newton’s classification of cubic curves. Illustrative examples of all the different possibilities are given. The results obtained extend to Krein spaces results of Kippenhahn on the classical numerical range.

It is shown that the asymptotic stability of positive 2D linear systems with delays is independent of the number and values of the delays and it depends only on the sum of the system matrices, and that the checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to testing that of the corresponding positive 1D systems without delays. The effectiveness of the proposed approaches is demonstrated on numerical examples.

For commuting elements x, y of a unital Banach algebra ℬ it is clear that $\parallel e^{x+y}\parallel \le \parallel e^x\parallel \parallel e^y\parallel$. On the order hand, M. Taylor has shown that this inequality remains valid for a self-adjoint operator x and a skew-adjoint operator y, without the assumption that they commute. In this paper we obtain similar inequalities under conditions that lie between these extremes. The inequalities are used to deduce growth estimates of the form $\parallel e^{\text{'}}\parallel \le c(1+|\xi |{⟩}^s$ for all $\xi \in R^m$, where $x=(x_1,...,x_m)\in {ℬ}^m$ and c, s are constants.

Let $a=(a_1,a_2,...,a_n)$ be a nonincreasing sequence of positive real numbers. Denote by $S=\{1,2,...,n\}$ the index set and by $J_k=\{I=\{r_1,r_2,...,r_k\}$, $1\le r_1<r_2<\cdots <r_k\le n\}$ the set of all subsets of $S$ of cardinality $k$, $1\le k\le n-1$. In addition, denote by $a_I=a_{r_1}+a_{r_2}+\cdots +a_{r_k}$, $1\le k\le n-1$, $1\le r_1<r_2<\cdots <r_k\le n$, the sum of $k$ arbitrary elements of sequence $a$, where $a_{I_1}=a_1+a_2+\cdots +a_k$ and $a_{I_n}=a_{n-k+1}+a_{n-k+2}+\cdots +a_n$. We consider bounds of the quantities $R{S}_k\left(a\right)=a_{I_1}/a_{I_n}$, $L{S}_k\left(a\right)=a_{I_1}-a_{I_n}$ and $S_{k,\alpha }\left(a\right)={\sum }_{I\in J_k}{a}_I^{\alpha }$ in terms of $A={\sum }_{i=1}^n{a}_i$ and $B={\sum }_{i=1}^n{a}_i^2$. Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

Quadratic forms on a free finite-dimensional module are investigated. It is shown that the inertial law can be suitably generalized provided the vector space is replaced by a free finite-dimensional module over a certain linear algebra over $ℝ$ ( real plural algebra) introduced in [1].

In this paper the problem of construction of the canonical matrix belonging to symplectic forms on a module over the so called plural algebra (introduced in [5]) is solved.