In 1941 Sumner Myers proved that if the Ricci curvature of a complete Riemann manifold has a positive infimum then the manifold is compact and its diameter is bounded in terms of the infimum. Subsequently the curvature hypothesis has been weakened, and in this paper we weaken it further in an attempt to find the ultimate, sharp result.

Some robotic applications like vacuum cleaners require the robot to travel to each and every spot in the room while avoiding obstacles whose position may or may not be known. These obstacles may not always be in the same position. The robot itself may or may not have a map of the room. In the latter case, it has to rely on its sensors to navigate. It also may not know its absolute position in t...

For resolving the problem of sprinkle repeated, overtaken and went beyond in irrigation, it is important to research the approach of irregular boundary area. The equation of range and flow for achieving the square and triangle spray were deduced. Pressure is proportional to range. Specific method of changing the sectional area was put forward for achieving square and triangle spray. Adopted MAT...

the aim of this paper is to present a numerical method for singularly perturbed convection-diffusion problems with a delay. the method is a combination of the asymptotic expansion technique and the reproducing kernel method (rkm). first an asymptotic expansion for the solution of the given singularly perturbed delayed boundary value problem is constructed. then the reduced regular delayed diffe...

We define the similarity boundary of a self-similar set and use it to analyze the properties of self-similar sets in the general setting of any complete metric space. The similarity boundary is an attempt at extending the concept of the topological boundary in a way that is consistent with our intuitive understanding of the term boundary. We also show how with the analysis of the similarity bou...

We investigate a relationship between the number of the negative modes around periodic instanton solution and the type of the decay-rate transition. It is shown that for the case of first-order decay-rate transition the lowest positive mode at low energy periodic instanton becomes additional negative mode at high energy regime, while in the second-order case there is only one negative mode in t...

We find the spectrum of boundary bound states for the sine-Gordon model with Dirichlet boundary conditions, closing the bootstrap and providing a complete description of all the poles in the boundary reflection factors. The boundary Coleman-Thun mechanism plays an important role in the analysis. Two basic lemmas are introduced which should hold for any 1+1-dimensional boundary field theory, all...

In this paper we describe all rotation H-hypersurfaces in Hn × R and use them as barriers to prove existence and characterization of certain vertical H-graphs and to give symmetry and uniqueness results for compact H-hypersurfaces whose boundary is one or two parallel submanifolds in slices. We also describe examples of translation H-hypersurfaces in Hn × R. For n > 2 we obtain a complete embed...