In this paper the problem of system equivalence is tackled for a rather general class of linear time-invariant systems. We consider AR-systems described by linear continuous shift-invariant operators with finite memory, acting on Frechet-signal spaces, containing the space £(lR) of infinitely differentiable functions on lR. This class is in one-one correspondence with matrices of suitable sizes...

We establish some versions of fixed-point theorem in a Frechet topological vector space E. The main result is that every map A BC where B is a continuous map and C is a continuous linear weakly compact operator from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Kra...

We investigate the maximal numberMk of offspring amongst all individuals in a critical Galton–Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index −α for 1 < α < 2, then k−1Mk converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

Improving a result of M. Rabus we force a normal, locally compact, 0-dimensional, Frechet-Uryson, initially ω1-compact and non-compact space X of size ω2 having the following property: for every open (or closed) set A in X we have |A| ≤ ω1 or |X \A| ≤ ω1.

For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties.We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous  differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.

Spaces of all entire functions f represented by vector valued Dirichlet series and having slow growth have been considered. These are endowed with a certain topology under which they become a Frechet space. On this space the form of linear continuous transformations is characterized. Proper bases have also been characterized in terms of growth parameters.