In this paper, we first give a characterization of sub-topical functions with respect to their lower level sets and epigraph. Next, by using two different classes of elementary functions, we present a characterization of sub-topical functions with respect to their polar functions, and investigate the relation between polar functions and support sets of this class of functions. Finally, we obtai...

The description of the error dynamics (30) in our paper [1] contains an omission that leads to some bounds used in the conditions of Theorem 8 and Corollary 2 in the paper to be incorrectly defined. In what follows, the correct error dynamics and the corresponding conditions are given. Instead of (30) in [1], the error dynamics are actually given by ė = m ∑ i=1 wi(z)[(Ai − LiC +MiAδi)e +Mi(Āδix̂...

Integrity for crisp graph theory is a well-defined topic. However, the integrity concept fuzzy graphs has only recently been defined and extensively researched. in m-polar (mPFG), each node as well edges m components. So, defining mPF environment needs new concept. As environment, edge components, so we have more flexibility to address uncertainty rather than other uncertain environments. In th...

Fuzzy models are present everywhere from natural to artificial structures, embodying the dynamic processes in physical, biological, and social systems. As real-life problems often uncertain on account of inconsistent indeterminate information, it seems very demanding for an expert solve those using a fuzzy model. In this regard, we develop hybrid new model m-polar Diophantine neutrosophic N-sof...

This paper is a brief summary of “M.Takahashi, T.Murofushi, S.Asahina, A new necessary and sufficient condition for the Egoroff theorem in non-additive measure theory, Fuzzy Sets and Systems, to appear.” This paper states that a newly defined condition, called condition (M), is a necessary and sufficient condition for the Egoroff theorem in non-additive measure theory. The existing necessary an...

The concept of upper and lower semicontinuity of fuzzy mappings introduced by Bao and Wu [Y.E. Bao, C.X. Wu, Convexity and semicontinuity of fuzzy mappings, Comput. Math. Appl., 51 (2006) 1809–1816] is redefined by using the concept of parameterized triples of fuzzy numbers. On the basis of the linear ordering of fuzzy numbers proposed by Goetschel and Voxman [R. Goetschel, W. Voxman, Elementar...

In (Fuzzy Sets and Systems 97 (1998) 33), we presented a fuzzy multipurpose decision making model integrating di4erent preference representations based on additive reciprocal fuzzy preference relations. The main aim of this paper is to complete the decision model studying conditions under which reciprocity property is maintained when aggregating preference relations using an OWA operator guided...