Let X,Y be normed spaces. The set of bounded linear operators is noted as L(X,Y ). Let now D = D(A) ⊂ X be a linear subspace, and A : D −→ Y a linear (not necessarily bounded!) operator. Notation: (A,D(A)) : X −→ Y Definition: G(A) := {(x,Ax) |x ∈ D} is called the graph of A. Obviously, G(A) is a linear subspace of X × Y . The linear operator A is called closed if G(A) is closed in X × Y . The ...

We present formulae relating controllability, observability, and co-observability arising in the context of supervisory control of discrete event systems. Given a discrete event plant G with event set Σ, uncontrollable event set Σui, and observation mask Mi of the ith supervisor, we first show that the infimal prefix-closed, (L(G), ⋂ i Σui)controllable, and (L(G),Σui,Mi)–co-observable superlang...

A graph G is arbitrarily decomposable into closed trails (ADCT) if the following is true: Whenever (l1, . . . , lp) is a sequence of integers adding up to |E(G)| and there is a closed trail of length li in G for i = 1, . . . , p, then there is a sequence (T1, . . . , Tp) of pairwise edge-disjoint closed trails in G such that Ti is of length li for i = 1, . . . , p. In the paper it is proved tha...

Let α(G), α(G), κ(G) and κ (G) denote the independence number, the matching number, connectivity and edge connectivity of a graph G, respectively. We determine the finite graph families F1 and F2 such that each of the following holds. (i) If a connected graph G satisfies κ (G) ≥ α(G) − 1, then G has a spanning closed trail if and only if G is not contractible to a member of F1. (ii) If κ (G) ≥ ...

Several specific types of generalized sets (briefly, g-T_g-sets in topological spaces T_g-spaces have been defined and investigated for various purposes from time to the literature T_g-spaces. Our recent research field a new class is reported herein as starting point more classes. It shown that superclass those whose elements are called open, closed, semi-open, semi-closed, pre-open, pre-closed...

Free-minor closed classes [2] and free-planar graphs [3] are considered. Versions of Kuratowski-like theorem for free-planar graphs and Kuratowski theorem for planar graphs are considered. We are using usual definitions of the graph theory [1]. Considering graph topologically and Kuratowski theorem, we use the notion of minor following the theory of Robertson and Seymour[2]. We say, that a grap...

In work spread over several decades, Dynkin ([4, 3]), Seitz ([10, 11]), and Testerman ([16]) classified the maximal closed connected subgroups of simple algebraic groups. Their analyses for the classical group cases were based primarily on a striking result: If G is a simple algebraic group and φ : G SL V is a tensor indecomposable irreducible rational representation, then with specified except...

A method for generating an interpolation closed G arc spline on a given closed point set is presented. For the odd case, i.e. when the number of the given points is odd, this paper disproves the traditional opinion that there is only one closed G arc spline interpolating the given points. In fact, the number of the resultant closed G arc splines fulfilling the interpolation condition for the od...

If G is a finitely presented group, then H2 (∞) (G,A) vanishes for all injective Banach spaces iff the regularized homological area function satisfies the linear isoperimetric inequality. This contrasts with the known result that G is word hyperbolic iff the homological area function satisfies the linear isoperimetric inequality. A closed 3-manifold group G is hyperbolic iff H2 (∞) (G,A) vanish...