A study of a general system of operator equations in $${\varvec{}}$$-metric spaces via the vector approach in fixed point theory

In this paper, we will present some coupled fixed point theorems on a metric space endowed with two b-metrics. The approach is based on the application of a fixed point theorem for an appropriate operator on the Cartesian product of the given spaces. An application to a boundary value problem for a system of second order differential equations is also presented.     

Four new topological indices based on the molecular path code

The sequence of all paths pi of lengths i = 1 to the maximum possible length in a hydrogen-depleted molecular graph (which sequence is also called the molecular path code) contains significant information on the molecular topology, and as such it is a reasonable choice to be selected as the basis of topological indices (TIs). Four new (or five partly new) TIs with progressively improved performance (judged by correctly reflecting branching, centricity, and cyclicity of graphs, ordering of alkanes, and low degeneracy) have been explored. (i) By summing the squares of all numbers in the sequence one obtains Sigmaipi(2), and by dividing this sum by one plus the cyclomatic number, a Quadratic TI is obtained: Q = Sigmaipi(2)/(mu+1). (ii) On summing the Square roots of all numbers in the sequence one obtains Sigmaipi(1/2), and by dividing this sum by one plus the cyclomatic number, the TI denoted by S is obtained: S = Sigmaipi(1/2)/(mu+1). (iii) On dividing terms in this sum by the corresponding topological distances, one obtains the Distance-reduced index D = Sigmai{pi(1/2)/[i(mu+1)]}. Two similar formulas define the next two indices, the first one with no square roots: (iv) distance-Attenuated index: A = Sigmai{pi/[i(mu + 1)]}; and (v) the last TI with two square roots: Path-count index: P = Sigmai{pi(1/2)/[i(1/2)(mu + 1)]}. These five TIs are compared for their degeneracy, ordering of alkanes, and performance in QSPR (for all alkanes with 3-12 carbon atoms and for all possible chemical cyclic or acyclic graphs with 4-6 carbon atoms) in correlations with six physical properties and one chemical property.     

Bernstein-type operators on a triangle with all curved sides

We construct and analyze Bernstein-type operators on triangles with curved sides, their product and Boolean sum. We study the interpolation properties and approximation accuracy. Using the modulus of continuity we also study the remainders of the corresponding approximation formulas. Finally, there are given some particular cases and numerical examples.     

Competition between Birch reduction and fluorine abstraction in reactions of hydrated electrons (H2O)n(-) with the isomers of di- and trifluorobenzene

The reactions of the isomers of di- and trifluorobenzene with hydrated electrons (H(2)O)(n)(-), n = 19-70, have been studied by Fourier transform ion cyclotron resonance (FT-ICR) mass spectrometry. While Birch reduction, i.e. H atom transfer to the aromatic ring, was observed for all studied isomers, a strong dependence on the substitution pattern was observed for fluorine abstraction. Nanocalorimetry combined with G3 calculations are used to analyze the thermochemistry of the reactions. Fluorine abstraction is at least 100 kJ mol(-1) more exothermic than Birch reduction, yet the latter is the dominant reaction pathway for all three difluorobenzene isomers. Fluorine abstraction and Birch reduction face activation barriers of comparable magnitude. The relative barrier height is sensitive to the substitution pattern. Birch reduction occurs selectively with 1,3- and 1,4-difluorobenzene in a nanoscale aqueous environment.     

Identifying Codes in Line Graphs

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbors within the code. We study the edge‐identifying code problem, i.e. the identifying code problem in line graphs. If denotes the size of a minimum identifying code of an identifiable graph G, we show that the usual bound , where n denotes the order of G, can be improved to in the class of line graphs. Moreover, this bound is tight. We also prove that the upper bound , where is the line graph of G, holds (with two exceptions). This implies that a conjecture of R. Klasing, A. Kosowski, A. Raspaud, and the first author holds for a subclass of line graphs. Finally, we show that the edge‐identifying code problem is NP‐complete, even for the class of planar bipartite graphs of maximum degree 3 and arbitrarily large girth.     

MPQ-trees for the Orthogonal Packing Problem

Given a set of rectangular items of different sizes and a rectangular container, the aim of the bi-dimensional Orthogonal Packing Problem (OPP-2 for short) is to decide whether there exists a non-overlapping packing of the items in this container. The rotation of items is not allowed. In this paper we present a new exact algorithm for solving OPP-2, based upon the characterization of solutions using interval graphs proposed by Fekete and Schepers. The algorithm uses MPQ-trees, which were introduced by Korte and M?hring to recognize interval graphs.     

Bernstein-type Operators on a Triangle with One Curved Side

We construct Bernstein-type operators on a triangle with one curved side. We study univariate operators, their product and Boolean sum, as well as their interpolation properties, the order of accuracy (degree of exactness, precision set) and the remainder of the corresponding approximation formulas. We also give some illustrative examples.