Let G be a simple graph and k\in\mathbb{N} . Two important graph operations are: the k -th graph power of G , denoted by G^{k} , where G^{k} is the graph obtained from G by adding an edge between every pair of vertices that have a distance at most k , and the complement graph of G , denoted by \overline{G} . In this paper, we studied the relation between the independence numbers of \overline{G}^{k} and \overline{G^k} .

We give a comprehensive account of zero mean curvature surfaces with planar curvature lines in isotropic 3 -space. After providing a complete classification of all such surfaces, we show that they belong to a 1 -parameter family of surfaces. We then investigate their relationship to Thomsen-type surfaces in isotropic 3 -space, those zero mean curvature surfaces in isotropic 3 -space that are also affine minimal.

In this paper, we use proof mining methods to compute rates of ( T -)asymptotic regularity of the generalized Krasnoselskii–Mann-type iteration associated to a nonexpansive mapping T:X\to X in a uniformly convex normed space X . For special choices of the parameter sequences, we obtain quadratic rates.

Let q_{1}, \ldots , q_{t} be distinct prime numbers. Let a_{1}, \ldots , a_{t} be nonnegative integers and x a positive integer. We establish an effective lower bound for the greatest prime divisor of |x^{2} - q_{1}^{a_1}\cdots q_{t}^{a_t}| , which tends to infinity with the maximum of x , a_{1}, \ldots , a_{t} .

This paper establishes topological semi-conjugacy between a symbolic dynamical system (\Sigma_{A},\sigma_{A}) and the system (M(X),T_{M}) induced on the space of probability measures by a compact topological dynamical system (X, T ) . Some conditions on an original system (X,T) are obtained for the induced system (M(X),T_M) to have a subsystem topologically semi-conjugate to (\Sigma_{A},\sigma_{A}) , and some relations between an original system (X,T) and the induced system (M(X),T_{M}) concerning semi-conjugacy to (\Sigma_{A},\sigma_{A}) are given. By using these results, several criteria of Li–Yorke chaos and Devaney chaos for the induced system (M(X),T_{M}) are established. Also, a characterization of the \mathcal{F} -mixing dynamical system (X,T) with Furstenberg family \mathcal{F} is obtained by means of the system (M(X),T_{M}) , which gives an answer to the question posed by Fu and Xing [Chaos Solitons Fractals (2012), 439–443].

We consider the fractional Schrödinger equation with a logarithmic nonlinearity, when the power of the Laplacian is between zero and one. We prove global existence results in three different functional spaces: the Sobolev space corresponding to the quadratic form domain of the fractional Laplacian, the energy space, and a space contained in the operator domain of the fractional Laplacian. For this last case, a finite momentum assumption is made, and the key step consists in estimating the Lie commutator between the fractional Laplacian and the multiplication by a monomial.

Regular abelian semigroups are isomorphic to a direct product of an abelian group and a rectangular band (Warne, 1994). Seeking for a similar result for nilpotency, solvability, and supernilpotency of regular semigroups, we obtain that an analogous statement is true only in orthodox semigroups. We provide an example that shows that the same does not have to be true in regular semigroups that are not orthodox.

This paper is dedicated to solving a multi-objective control problem for a 2\times2 system of parabolic equations. Here, we have many objectives, possibly conflictive, and we adopt a concept of hierarchy for the controls. We have the leader control, responsible for objectives of controllability type, and the follower controls which intend to be a Nash-equilibrium with respect to some given cost functionals. The novelty here is that we formulate this problem for systems of parabolic equations, meaning that we have many variables and naturally much more objectives to accomplish.

Recently, we have studied the relation between the singular spectrum of a class of tempered ultrahyperfunctions corresponding to proper convex cones and their expressions as boundary values of holomorphic functions. In this note, we added some more results on the singular spectrum of the tempered ultrahyperfunctions. More specifically, the “microlocalization” of a Silva–Hasumi–Morimoto ultrahyperfunction corresponding to an open proper convex cone is addressed. Following Nishiwada’s approach to the analytic wave front set, we introduce the real singular spectrum of such an ultrahyperfunction and prove that it canonically projects onto Silva’s real support.

The aim of this paper is to study the obstacle problem associated with noncoercive elliptic equations with variable exponents and L^1 -data. After proving the existence and regularity of entropy solutions, we have extended the Lewy–Stampacchia inequalities to the case of noncoercive elliptic operators.