A new class of fractional linear continuous-time linear systems described by state equations is introduced. The solution to the state equations is derived using the Laplace transform. Necessary and sufficient conditions are established for the internal and external positivity of fractional systems. Sufficient conditions are given for the reachability of fractional positive systems.

The error analysis of preconditioned waveform relaxation iterations for differential systems is presented. This analysis extends and refines previous results by Burrage, Jackiewicz, Nørsett and Renaut by incorporating all terms in the expansion of the error of waveform relaxation iterations in the Laplace transform domain. Lower bounds for the size of the window of rapid convergence are also obtained. The theory is illustrated for waveform relaxation methods applied to differential systems resulting...

Suppose that the function $q\left(t\right)$ in the differential equation (1) $y^{\text{'}\text{'}}+q\left(t\right)y=0$ is decreasing on $(b,\infty )$ where $b\ge 0$. We give conditions on $q$ which ensure that (1) has a pair of solutions $y_1\left(t\right),y_2\left(t\right)$ such that the $n$-th derivative ($n\ge 1$) of the function $p\left(t\right)=y_1^2\left(t\right)+y_2^2\left(t\right)$ has the sign ${(-1)}^{n+1}$ for sufficiently large $t$ and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.

We consider the Gaudin model associated to a point z ∈ ℂⁿ with pairwise distinct coordinates and to the subspace of singular vectors of a given weight in the tensor product of irreducible finite-dimensional sl₂-representations, [G]. The Bethe equations of this model provide the critical point system of a remarkable rational symmetric function. Any critical orbit determines a common eigenvector of the Gaudin hamiltonians called a Bethe vector. In [ReV], it was shown that for generic...

Definitions, properties, examples and applications of generalized analytic functions introduced by B. Ziemian are presented.

Let $L\left(y\right)=y^{\left(n\right)}+q_{n-1}\left(t\right)y^{(n-1)}+\cdots +q_0\left(t\right)y,t\in [a,b)$, be an $n$-th order differential operator, $L^{*}$ be its adjoint and $p,w$ be positive functions. It is proved that the self-adjoint equation $L^{*}\left(p\left(t\right)L\left(y\right)\right)=w\left(t\right)y$ is nonoscillatory at $b$ if and only if the equation $L\left(w^{-1}\left(t\right)L^{*}\left(y\right)\right)=p^{-1}\left(t\right)y$ is nonoscillatory at $b$. Using this result a new necessary condition for property BD of the self-adjoint differential operators with middle terms is obtained.

In this paper first order systems of linear of ODEs are considered. It is shown that these systems admit unique solutions in the Colombeau algebra $ℒ\left({𝐑}^1\right)$.