A perfect independent set I of a graph G is defined to be an independent set with the property that any vertex not in I has at least two neighbors in I. For a nonnegative integer k, a subset I of the vertex set V (G) of a graph G is said to be k-independent, if I is independent and every independent subset I' of G with |I' | ≥ |I| − (k − 1) is a subset of I. A set I of vertices of G is a super k-independent set of G if I is k-independent in the graph G[I, V (G) − I], where G[I, V (G) − I] is the bipartite graph obtained from G by deleting all edges which are not incident with vertices of I. It is easy to see that a set I is 0-independent if and only if it is a maximum independent set and 1-independent if and only if it is a unique maximum independent set of G. In this paper we mainly investigate connections between perfect independent sets and k-independent as well as super k-independent sets for k = 0 and k = 1.
Assuming that $(\Omega , \Sigma , \mu )$ is a complete probability space and $X$ a Banach space, in this paper we investigate the problem of the $X$-inheritance of certain copies of $c_0$ or $\ell _{\infty }$ in the linear space of all [classes of] $X$-valued $\mu $-weakly measurable Pettis integrable functions equipped with the usual semivariation norm.
In this paper we extend our results given in [5] where we compared PEG systems with pure regulated context-free grammars (see [3]). We will show that the family of languages generated by the pure grammars of type 0 is a proper subclass of the family of languages generated by positioned eco-grammar systems. We present a way how to coordinate parallel behavior of agents with one-sided context in a PEG system in order to simulate the derivation step in a pure grammar of type 0 determined by a single rule which replaces an arbitrarily long string by another one. Related results concerning PEG systems and pure languages can be found in [6].
In this paper we study nonlinear second order differential equations subject to separated linear boundary conditions and to linear impulse conditions. Sign properties of an associated Green’s function are investigated and existence results for positive solutions of the nonlinear boundary value problem with impulse are established. Upper and lower bounds for positive solutions are also given.
A possibilistic marginal problem is introduced in a way analogous to probabilistic framework, to address the question of whether or not a common extension exists for a given set of marginal distributions. Similarities and differences between possibilistic and probabilistic marginal problems will be demonstrated, concerning necessary condition and sets of all solutions. The operators of composition will be recalled and we will show how to use them for finding a T-product extension. Finally, a necessary and sufficient condition for the existence of a solution will be presented.
There exist different formulations of the irreversible thermodynamics. Depending on the distance from the equilibrium state and on the characteristic time the main theories are the classical theory (CIT), the thermodynamics with internal variables (IVT) and the extended theory (EIT). Sometimes it is not easy to choose the proper theory and to use it efficiently with respect to applied problems considering different fields of interest. Especially EIT is explained mainly for very special choice of the dissipative fluxes under specific presumptions. The paper tries to formulate EIT and IVT in a simple, unified but general enough form. The basic presumptions for EIT are shown and discussed, further a possible generalization is proposed. The formulation allows the integration of iVT and EIT even for the mixture of chemically interacting components and diffusion. The application of the formulation is demonstrated on an example. and Obsahuje seznam literatury
For given a graph $H$, a graphic sequence $\pi =(d_1,d_2,\ldots ,d_n)$ is said to be potentially $H$-graphic if there is a realization of $\pi $ containing $H$ as a subgraph. In this paper, we characterize the potentially $(K_5-e)$-positive graphic sequences and give two simple necessary and sufficient conditions for a positive graphic sequence $\pi $ to be potentially $K_5$-graphic, where $K_r$ is a complete graph on $r$ vertices and $K_r-e$ is a graph obtained from $K_r$ by deleting one edge. Moreover, we also give a simple necessary and sufficient condition for a positive graphic sequence $\pi $ to be potentially $K_6$-graphic.