In this chapter we introduce the fundamental concepts behind the Green Function Theory. We start from a description of the second quantization method and then, after introducing the ensemble average of an operator, we define the main object of the theory, the Green function. Further on, after obtaining the equation of motion for the Green function, we introduce the self-energy operator and we d...

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green’s formula, the relation between integral and differential forms, conservation laws, ...

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary rst step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formula, the connection between integral and di erential forms, conservation laws, E...

In this paper, we prove that every multiplier M (i.e. every bounded operator commuting whit the shift operator S) on a large class of Banach spaces of sequences on Z is associated to a function essentially bounded by ‖M‖ on spec(S). This function is holomorphic on ◦ spec(S), if ◦ spec(S) 6= ∅. Moreover, we give a simple description of spec(S). We also obtain similar results for Toeplitz operato...

We study systematically the Lax description of the KdV hierarchy in terms of an operator which is the geometrical recursion operator. We formulate the Lax equation for the n-th flow, construct the Hamiltonians which lead to commuting flows. In this formulation, the recursion relation between the conserved quantities follows naturally. We give a simple and compact definition of all the Hamiltoni...

Direct and inverse scattering problem for an operator with non-local potential is solved in the paper. The method based on Riemann boundary value a bundle of three straight lines. Description data given.

Computing least common subsumers in description logics is an important reasoning service useful for a number of applications. As shown in the literature, this reasoning service can be used for the approximation of concept disjunctions in description logics, the \bottom-up" construction of knowledge bases, learning tasks, and for speciic kinds of information retrieval. So far, computing the leas...

We present a nondeterministic polynomial-time decision procedure for an extension of multi-level syllogistic with the singleton operator, the Cartesian product, various map constructs, and special predicates asserting respectively that a map term is single valued, injective, and bijective, but with no map image or domain operators. We also prove that the extension of multi-level syllogistic wit...

Given a description of the quantitative information flow (qif) for components, how can we determine the qif of a system composed from components? We explore this fundamental question mathematically and provide an answer based on a new composition operator. We investigate its properties and prove that it generalises existing composition operators. We illustrate the results with a fresh look on C...

We study the resonant set of a two-level Schrödinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr...