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.

S U M M A R Y We present an alternative to the classical mode coupling method scheme often used in global seismology to compute synthetic seismograms in laterally heterogeneous earth model and Frechet derivatives for tomographic inverse problem with the normal modes first-order Born approximation. We start from the first-order Born solution in the frequency domain and we use a numerical scheme ...