We show that if two 3-manifolds with toroidal boundary are glued via a “sufficiently complicated” map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, suppose X∪F Y is a manifold obtained by gluing X and Y , two connected small manifolds with incompressible boundary, along a closed surface F . Then the following inequality on genera is obtained:

This paper investigates the appearance manifold of facial expression: embedding image sequences of facial expression from the high dimensional appearance feature space to a low dimensional manifold. We explore Locality Preserving Projections (LPP) to learn expression manifolds from two kinds of feature space: raw image data and Local Binary Patterns (LBP). For manifolds of different subjects, w...

It is shown in this paper that if BG is the Stone-ech compactification of a group G, and G satisfying a certain condition, then there is a weakly recurrent point in BG which is not almost periodic, and if another condition will be added, then there is a recurrent point in 8G which is not almost periodic point.

The purpose of this paper is to introduce and study the concepts of intuitionistic fuzzy almost weakly generalized closed mappings and intuitionistic fuzzy almost weakly generalized open mappings in intuitionistic fuzzy topological space. Some of their properties are explored.