Volume 62, Issue 4

The eternal domination problem requires a graph be protected against an infinitely long sequence of attacks at vertices, by guards located at vertices, with the requirement that the configuration of guards induces a dominating set at all times. An attack is defended by sending a guard from a neighboring vertex to the attacked vertex. We allow all guards to move to neighboring vertices in response to an attack, but allow the attacked vertex to choose which neighboring guard moves to the attacked vertex. This is the all-guards move version of the “foolproof” eternal domination problem that has been previously studied. We present some results and conjectures on this problem.

Weak effect algebras were introduced by the author as a generalization of effect algebras and pseudoeffect algebras. It was shown that having a basic algebra, we can restrict its binary operation to orthogonal elements only and what we get is just a weak effect algebra. However, the converse construction is impossible due to the fact that the underlying poset of a basic algebra is a lattice which need not be true for weak effect algebras. Hence, we found a weaker structure than a basic algebra which can serve as a representation of a weak effect algebra.

The dual of the join semilattice of proper compact Scott open subsets of a domain D is called the Smyth powerdomain of D. The Smyth powerdomain is used in programming semantics as a model for demonic nondeterminism. In this paper, we introduce the concept of partial information systems; and, as an application, show that the Smyth powerdomain of any domain can be realized in terms of the sub partial information systems of the domain’s corresponding information system.

We introduce the notion of torsion class of abelian cyclically ordered groups; the definition is analogous to that used in the theory of lattice ordered groups. The collection T of all such classes is partially ordered by the class-theoretical inclusion. Though T is a proper class, we can apply the usual terminology for this partial order. We prove that T is a complete, infinitely distributive lattice having infinitely many atoms.

Using methods of categorical fuzzy topology, the paper shows a relation between topological systems of S. Vickers and Artin glueing of M. Artin. Inspired by the problem of interrelations between algebra and topology, we show the necessary and sufficient conditions for the category, obtained by Artin glueing along an adjoint functor, to be (co)algebraic and (co)monadic, incorporating the respective result of G. Wraith. As a result, we confirm the algebraic nature of the category of topological systems, showing that it is monadic.

By combining a telescopic summation formula with Kummer-Thomae-Whipple transformation, we prove two nonterminating 3 F 2(1)-series identities with one of them confirming a conjecture by Milgram (2009) and another one extending a couple of terminating series identities due to Gessel and Stanton (1982).

The existence of anti-periodic solutions of the following nonlinear impulsive functional differential equations $$ x'(t) + a(t)x(t) = f(t,x(t),x(\alpha _1 (t)), \ldots ,x(\alpha _n (t))),t \in \mathbb{R},\Delta x(t_k ) = I_k (x(t_k )),k \in \mathbb{Z} $$ is studied. Sufficient conditions for the existence of at least one anti-periodic solution of the mentioned equation are established. Several new existence results are obtained.

The geometric character of domains of solutions of a singular parabolic equation with the Neumann boundary condition are discussed in this work. We prove a gradient estimate and a Harnack type inequality and then, show that the domains of the solutions are exactly columns in the space-time. Specially, the altitudes of the columns are calculated accurately.

Elementary version of moving frame method is applied to the symmetries and equivalences of second-order difference equations. Close links to the web theory and Riemannian geometry are established.

The paper deals with the questions: (a)whether a topological module admits maximal linearly independent subsets that are analytic(b)whether an Abelian topological group admits maximal independent subsets that are analytic(c)whether a topological field extension admits transcendence bases that are analytic.

An uncertainty of an estimated parameter can be, in general, decomposed into two parts, i.e. an uncertainty caused by errors in the actual experiment (the type A uncertainty) and an uncertainty caused by errors in preceding experiment, where some wanted constants were estimated (the type B uncertainty). These constants are necessary for an estimation of the useful parameters. The aim of the paper is to find a condition for elimination of the type B uncertainty.

Let in a linear model with large number of parameters some parameters be neglected and remaining parameters be changed in such a way (if it is possible) that the new model still corresponds with data. In the new (reduced) model problems of confidence regions and testing statistical linear hypotheses are investigated.

A structure on terms of faster convergent series is studied in the paper. Necessary and sufficient conditions for the existence of faster convergent series with different types of terms are found. A faster convergence criteria for certain Kummer’s series is proved in this paper.

It is well known that the difference between the set of all ordinary density points and the set of all strong density points of an arbitrary measurable subset of the plane is a null set. It is of interest to check how large can a difference between sections of these sets be. Both measure and category cases are considered.