In section 1 we present definitions and basic results concerning the Mazur intersection property (MIP) and some of its related properties as the MIP* . Section 2 is devoted to renorming Banach spaces with MIP and MIP*. Section 3 deals with the connections between MIP, MIP* and differentiability of convex functions. In particular, we will focuss on Asplund and almost Asplund spaces. In Section 4...

We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner’s lemma. A notable application of this technique yields a cake-cutting theorem where the number of players and the number of pieces can be independently chosen. We also prove multilabeled versions of Fan’s lemma, a combinatorial analogue of the Borsuk-Ulam theorem, and ...

K e y w o r d s H y e r s U l a m stability, Flett mean value theorem, Flett's point. 1. I N T R O D U C T I O N In 1968, Ulam [1] proposed the general problem: "When is it t rue tha t by changing a l i t t le the hypotheses of a theorem one can still assert t ha t the thesis of the theorem remains t rue or approx imate ly t rue?" In 1978, Gruber [2] proposed the following Ulam type problem: "S...

Abstract In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on finite interval was considered. The term in the contains multiple time delays. addition to giving some new theorems existence and uniqueness of solutions equation, authors also prove Hyers-Ulam-Rassias stability Hyers-Ulam equation. proofs use several different tools including B...

The generalized Schoenflies theorem asserts that if φ ∶ Sn−1 → S is a topological embedding and A is the closure of a component of Sn∖φ(Sn−1), then A ≅ D as long as A is a manifold. This was originally proved by Barry Mazur and Morton Brown using rather different techniques. We give both of these proofs.

C-cross topologies are introduced. Modifications of the Kuratowski-Ulam Theorem are considered. Cardinal invariants add , cof, cov and non with respect to meager or nowhere dense subsets are compared. Remarks on invariants cof(nwdY ) are mentioned for dense subspaces Y ⊆ X.