We show that the class of level-k control languages, as defined by Weir, is properly included in the class of 2 k−1-multiple context-free languages for each k ≥ 2. The proof of inclusion uses a representation of the set of derivation trees for a level-k control language in terms of a second-order abstract categorial grammar. 1.1 Introduction Weir (1992a) introduced a hierarchy of language class...

In this study, generalized k-order Fibonacci hybrid quaternion is defined. We give recurrence relation, generating function, the summation formula and some properties for these quaternions. Furthermore, matrix representation quaternions determined. The Q_k defined numbers given By means of another matrices, several identities are also obtained.

We present a canonical derivation of an influence superoperator which generates the reduced dynamics Fermionic quantum system linearly coupled to environment initially at thermal equilibrium. use this formalism derive generalized-Lindblad master equation (in Markovian limit) and generalized version hierarchical equations motion valid in arbitrary parity-symmetry conditions, important for correc...

Introducing the notions of (inner) σ-derivation, (inner) σ-endomorphism and oneparameter group of σ-endomorphisms (σ-dynamics) on a Banach algebra, we correspond to each σ-dynamics a σ-derivation named as its σ-infinitesimal generator. We show that the σ-infinitesimal generator of a σ-dynamics of inner σ-endomorphisms is an inner σ-derivation and deal with the converse. We also establish a nice...

Let G be a simple connected graph. The generalized polarity Wiener index of G is defined as the number of unordered pairs of vertices of G whose distance is k. Some formulas are obtained for computing the generalized polarity Wiener index of the Cartesian product and the tensor product of graphs in this article.