We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear spaces. This resolves famous Tingley's problem in class

Two important theorems in algebraic topology are the Brouwer Fixed Point theorem and the Borsuk-Ulam theorem. The theorems require the development of homology in their standard proofs. However, each theorem has an equivalent combinatorial result involving triangulating the relevant surface and coloring the vertices of the triangulation. Then by taking the limit of a sequence of finer triangulat...

In 1952, Ky Fan proved a combinatorial theorem generalizing the Borsuk-Ulam theorem stating that there is no Z2-equivariant map from the d-dimensional sphere S to the (d − 1)-dimensional sphere Sd−1. The aim of the present paper is to provide the same kind of combinatorial theorem for Dold's theorem, which is a generalization of the Borsuk-Ulam theorem when Z2 is replaced by Zq, and the spheres...

In this paper, we investigate four different types of Ulam stability, i.e., Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for a class of nonlinear implicit fractional differential equations with non-instantaneous integral impulses and nonlinear integral boundary condition. We also establish certain conditions fo...

Recent advances in neuronal multisensory integration suggest that the five senses do not exist in isolation of each other. Perception, cognition and action are integrated at very early levels of central processing, in a densely-coupled system equipped with multisensory interactions occurring at all temporal and spatial stages. In such a novel framework, a concept from the far-flung branch of to...

We prove that every surjective isometry from the unit sphere of space K(H), all compact operators on an arbitrary complex Hilbert H, onto real Banach Y can be extended to a linear K(H) Y. This is probably first example infinite dimensional non-commutative C*-algebra containing no unitaries and satisfying Mazur-Ulam property. also C*-algebras weakly JB*-triples satisfy

The Sperner and Tucker lemmas are combinatorial analogous of the Brouwer and Borsuk Ulam theorems with many useful applications. These classic lemmas are concerning labellings of triangulated discs and spheres. In this paper we show that discs and spheres can be substituted by large classes of manifolds with or without boundary.