Basic sciences

In this paper, we introduce the  probabilistic normed groups. Among other results, we investigate the continuityof inner automorphisms of a group and the continuity of left and right shifts in probabilistic group-norm. We also study midconvex functions defined  on probabilistic normed groups and  give  some results about locally boundedness of such  functions.

In this paper, first we define and study the probabilistic n-normed spaces and D-n-compactness, also we prove some theorems and inequalities. In the next section we define D-n-boundedness and prove some results in relation between D-n-compact and D-n-bounded sets in these spaces.

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

Abstract. K. Menger introduced the notion of probabilistic metric spaces. The idea of Menger was to use distribution function instead of nonnegative real numbers as values of the metric. The concept of probabilistic normed spaces were introduced by Serstnev. Then I. Golet defined generalized probabilistic 2-normed space (briefly GP-2-N space). In this paper, we defined bounded sets in GP-2-N space and we studied the relationship among D-bounded sets and proved some result in this space.

In this paper, first we define and study the probabilistic n-normed spaces and Dn-compactness, also we prove some theorems and inequalities. In the next section we define Dn-boundedness and prove some results in relation between Dn-compact and Dn-bounded sets in these spaces.

The purpose of this paper is to prove some common fixed point theorems for a pair of R-weakly commuting mappings defined on intuitionistic fuzzy metric spaces [Park JH. Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals 2004;22:1039–46] and L-fuzzy metric spaces [Saadati R, Razani A, Adibi H. A common fixed point theorem in L-fuzzy metric spaces. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2006.01.023], with nonlinear contractive condition, defined with function, first observed by Boyd and Wong [Boyd DW, Wong JSW. On nonlinear contractions. Proc Am Math Soc 1969;20:458–64]. Following Pant [Pant RP. Common fixed points of noncommuting mappings. J Math Anal Appl 1994;188:436–40] we define R-weak commutativity for a pair of mappings and then prove the main results. These results generalize some known results due to Saadati et al., and Jungck [Jungck G. Commuting maps and fixed points. Am Math Mon 1976;83:261–3]. Some examples and comments according to the preceding results are given.

ABSTRACT The aim of this paper is to give necessary conditions to get bounded linear operators in random 2-normed linear spaces and to study the relation between the boundedness and continuity of a linear operator in random 2-normed linear spaces.

This paper examined the relationship between Mathematics anxiety, Mathematics performances and overall academic performance in high school students. 424 (111 males and 113 females) of high school students from three states in South India, responded to a Mathematics Anxiety Scale and Socio Demographic Questionnaire. The effects of gender on Mathematics anxiety, Mathematics performance and overall academic performance were examined and after data collection Pearson Correlation Analysis and two Independent Sample T test were used to analyze the data. Consequences reveal that Mathematics Anxiety significantly has negative correlation with Mathematics performances and overall academic performance. Moreover it was found that there is significant gender difference in Mathematics anxiety, whereas there is no significant difference between boys and girls in Mathematics performances and academic performance.

In this paper, first, using interpolative Kannan type contractions, a new fixed point theorem has been proved. Then, by applying sequentially convergent mappings and using subadditive altering distance functions, we generalize contractions in complete metric spaces. Finally, we investigate some fixed point theorems which are generalizations of Kannan and Reich fixed points.

In this paper, a D2D multiple-metric routing protocol is proposed in the situation that the network infrastructures are ruined due to a possible natural disaster. To improve the stability and the shortness of the path, we use fuzzy mathematics and probabilistic normed spaces in such a way that the set of nodes is considered as a fuzzy set. This protocol operates in a distributed manner and at each stage independently finds the appropriate node. By introducing a stability indicator, we show that our method, while close to the shortest path, maintains durability.

In this paper, a device-to-device (D2D) communication multiple-metric scenario for resource allocation and pairing is proposed. To improve the stability of the D2D connection, a fuzzy method based on the data-rate and battery levels of potential D2D pairs is studied so that the set of D2D transmitters is considered as a fuzzy set. A single-cell scenario with a cellular user and some co-channel D2D nodes consisting of D2D receivers and transmitters is considered. A stable fuzzy pairing criterion is proposed for the selection of the best D2D transmitter. The proposed method is compared to three other pairing methods: maximum sum-rate, constant, and random pairing methods. The simulation results show that the proposed pairing method outperforms the other three methods in terms of stability and fairness criteria and follows the maximum sum-rate method from the sum-rate criteria point of view.