This note aims to give an elementary proof for Toponogov’s theorem in Alexandrov geometry with lower curvature bound. The idea of the comes from fact that, Riemannian geometry, sectional can be embodied geodesic variations.

In this note, we continue the works in the paper [Some properties of L-fuzzy approximation spaces on bounded integral residuated lattices", Information Sciences, 278, 110-126, 2014]. For a complete involutive residuated lattice, we show that the L-fuzzy topologies generated by a reflexive and transitive L-relation satisfy (TC)L or (TC)R axioms and the L-relations induced by two L-fuzzy topologi...

We introduce the concept of the intuitionistic fuzzy proximity as a generalization of fuzzy proximity, and investigate its properties. Also we investigate the relationship among intu-itionistic fuzzy proximity and fuzzy proximity, and intuitionistic fuzzy topology. 1. Introduction. As a generalization of fuzzy sets, the concept of intuitionistic fuzzy sets was introduced by Atanassov [1]. Recen...

In this paper, we proved the Alexandrov-Fenchel inequalities for embedded, closed, connected and convex C 2 -hypersurfaces in S n + 1 : A k ≥ ξ , − ( ) any ≤ where is quermassintegral (see Definition 1.1) unique positive function such that equality holds when M a geodesic sphere.

We review some aspects of the geometry of length spaces and metric spaces, in particular Alexandrov spaces with curvature bounded below and/or above. We then point out some possible directions of research to explore connections between the synthetic approach to Riemannian geometry and some aspects of the approach to non-smooth differential geometry through generalised functions. AMS Mathematics...

In this paper, we investigate the properties of join preserving maps in complete residuated lattices. We define join approximation operators as a generalization of fuzzy rough sets in complete residuated lattices. Moreover, we investigate the relations between join preserving operators and Alexandrov fuzzy topologies. We give their examples. AMS Subject Classification: 03E72, 03G10, 06A15, 06F07