Narrow Results

The aim of the present work is twofold: to develop numerical procedures for a priori determining whether a given cell population, having a distributed cell-cycle duration, will grow or decay when subjected to prescribed chemotherapy; to evaluate the cumulative error in the long-term predictions for such populations. We show that cell population dynamics under drug treatment can be modelled by iterative application of a compact operator on the initial cell age-distribution. We further show that this model can be approximated by iterative application of matrices on some finite-dimensional vector, containing initial conditions. Moreover, we develop a method for estimating the growth rate of cell population and show that in fully periodic treatments the estimated error does not grow as time tends to infinity. From the biomedical viewpoint this means that only fully periodic (strictly periodic) schedules can be considered for successfully predicting the long-term effect of chemotherapy. Thus, cyclic drug treatment is shown to be advantageous, not only in increasing selectivity of chemotherapy, as has been previously demonstrated, but also in increasing long-term predictability of specific treatment schedules.

Let $A$ be a compact operator in a separable Hilbert space and ${\lambda }_k(A)(k=1,2,\dots )$ be the eigenvalues of $A$ with their multiplicities enumerated in the non-increasing order of their absolute values. We prove the inequality

In this paper we investigate the measurable transformations that induce composition operators on L^ϕ-1 spaces and study some properties of these operators.

In this article we first consider the composition operator T on certain Banach spaces of holomorphic functions on a domain V ⊂ Cⁿ, which gives us the "Schröder" equation. Then we will consider a general operator, namely, a weighted composition operator which induces a generalization of the "Schröder" equation. We will study the problem of eigennumbers (or eigenfunctions) of the operator T.

We describe the main interpolation methods for Banach couples, their roots and some of their properties and applications. We consider also factorization of weakly compact operators and interpolation of couples of Banach algebras.