The Hamiltonian approach of the Pauli-Fierz model is studied by Derezinski and Jaksic (cf. [7]). In this paper, we give the Markovian description of this model. Using the weak coupling limit, we derive the quantum Markovian semigroup of the Pauli-Fierz system from its Hamiltonian description. Also, we give the explicit form of the associated Lindblad generator. As a consequence, we study the pr...

In this paper, we frequently use the notation from [S1]. A “rational semigroup” G is a semigroup generated by non-constant rational maps g : CI → CI, where CI denotes the Riemann sphere, with the semigroup operation being functional composition. For a rational semigroup G, we set F (G) := {z ∈ CI | G is normal in a neighborhood of z} and J(G) := CI \ F (G). F (G) is called the Fatou set of G an...

In [1] we introduced the concept of restricted representations for an inverse semigroup S and studied the restricted forms of some important Banach algebras on S. In this paper, we continue our study by considering the relation between the restricted positive definite functions and retricted representations. In particular, we prove restricted versions of the Godement’s characterization of the p...

Nonlinear dynamical automata (NDAs) are implementations of Turing machines by nonlinear dynamical systems. In order to use them as parsers, the whole string to be processed has to be encoded in the initial conditions of the dynamics. This is, however, rather unnatural for modeling human language processing. I shall outline an extension of NDAs that is able to cope with that problem. The idea is...

Probability distributions which can be obtained from superpositions of Gaussian distributions of different variances v=sigma;{2} play a favored role in quantum theory and financial markets. Such superpositions need not necessarily obey the Chapman-Kolmogorov semigroup relation for Markovian processes because they may introduce memory effects. We derive the general form of the smearing distribut...

In this paper we use the description of free group factors as the von Neumann algebras of Berezin’s deformation of the upperhalf plane, modulo PSL2(Z). The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive — 2 Hochschild cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator, L, for...

Let S be a semigroup. The intersection graph of an S-act A, denoted by G(A), is the undirected simple graph obtained by setting all non-trivial subacts of A to be the vertices and defining two distinct vertices to be adjacent if and only if their intersection is non-empty. It is investigated the interplay between the algebraic properties of A and the graph-theoretic properties of G(A). Also som...

We consider the class of quantum mechanical master equations defined on a generic Banach space, arising by projecting weakly perturbed one-parameter groups of isometries. We show that the possible semigroup approximations are far from unique. However, uniqueness can be reestablished through the introduction of a dynamical time averaging map. The generator of the resulting Contraction Semigroup ...