The Nielsen-Schreier Theorem states that every subgroup of a free group is free. To formalize this theorem in weak subsystems of second order arithmetic, one has to choose between defining a subgroup in terms of a set of group elements and defining it in terms of a set of generators. We show that if subgroups are defined by sets, then the Nielsen-Schreier Theorem is provable in RCA0, while if s...

A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. Recently Andersen et al. [3] and Cornuéjols and Margot [17] showed that the facet-defining inequalities of this set are either split cuts or intersection cuts obtained from lattice-free triangles and quadrilaterals. Throug...

A graph is α-critical if its stability number increases whenever an edge is removed from its edge set. The class of α-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of Lovász (1978) is the finite basis theorem for α-critical graphs of a fixed def...

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...

In this paper the questions of defining the optimum allocation of centers in fuzzy transportation networks are observed by the minimax criterion. It is supposed that the information received from the geographical information system is presented as a fuzzy graph. In this case the task of defining optimum allocation of the centers transforms into the task of defining the fuzzy set of graph bases....

Characterization of the containment of a polyhedral set in a closed halfspace, a key factor in generating knowledge-based support vector machine classifiers [7], is extended to the following: (i) Containment of one polyhedral set in another. (ii) Containment of a polyhedral set in a reverse-convex set defined by convex quadratic constraints. (iii) Containment of a general closed convex set, def...

In this paper, by defining soft ditopological spaces, a new approach to the soft topology via softset theory is made, with defining two structures on a soft set a soft topology and a softsubspace topology. Moreover, separation axioms in soft ditopological spaces are characterizedand the relations between soft topological and soft ditopological structures are investigated.Referen...

In this paper, by defining soft ditopological spaces, a new approach to the soft topology via softset theory is made, with defining two structures on a soft set a soft topology and a softsubspace topology. Moreover, separation axioms in soft ditopological spaces are characterizedand the relations between soft topological and soft ditopological structures are investigated.Referen...