Let M be a von Neumann algebra and self-adjoint operator affiliated with M. We define the notion of an “integral symmetrically normed ideal” introduce space OC[k](R)⊆Ck(R) functions R→C such that following holds: for any integral ideal I f∈OC[k](R), function Isa∋b↦f(a+b)−f(a)∈I is k-times continuously Fréchet differentiable, formula its derivatives may written in terms multiple integrals. Moreo...

The Kuznetsov–Index of a modal logic is the least cardinal μ such that any consistent formula has a Kripke–model of size ≤ μ if it has a Kripke–model at all. The Kuznetsov–Spectrum is the set of all Kuznetsov–Indices of modal logics with countably many operators. It has been shown by Thomason that there are tense logics with Kuznetsov–Index iω+ω. Futhermore, Chagrov has constructed an extension...

Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduc...