It is shown that a soft set can be represented as a crisp set of soft elements and a soft group as a ordinary group of soft elements. From this view point it is immediate that soft group share the properties of ordinary group. Also using soft elements the definitions of soft co-sets, soft homomorphism and cyclic soft groups are presented and their properties are studied.
In this article, we introduce the concept of soft intervals, soft ordering and sequences of soft real numbers, and some of their structural properties are studied. The notion of soft Lebesgue measure on the soft real numbers has been introduced. Also, a correspondence relationship has been established between the soft Lebesgue measure and the classical Lebesgue measure. Furthermore, we have studied some exciting results and relations between the soft Lebesgue measure and the Lebesgue measure of soft real sets.
In this paper, the notion of soft [Formula: see text] on the soft sets has been introduced. A correspondence relationship between the soft [Formula: see text] and the [Formula: see text] has been established. Consequently, soft measure is defined over the soft [Formula: see text] and the relationship between the soft measure and the corresponding measure has been drawn. Finally, the soft outer measure on the soft power set is coined and also the correspondence between the soft outer measure and the measure has been depicted.
A delayed, discrete-time, prey-predator model with Allee effects imposed on prey and predator populations is defined, and dynamics of the system is characterized computationally. The parametric conditions for local asymptotic stability of equilibria are investigated with several illustrating examples. In addition, several periodic solutions were achieved. Furthermore, high degree of fractality of the prey-predator population trajectories is observed when the system is sufficiently delayed. This system has turned out to be a cooperative system, which is depicted using fractal dimension.
In this paper, we introduce the notion of the soft topological group as a general topological group of soft elements and study their properties. Some interesting results and relation between soft topological group and ordinary topological group of soft elements are studied. Also, soft Borel set and soft Borel measure have been depicted and the relationship between the soft Borel measure and the corresponding Borel measure has been drawn.
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