We present an analysis of bounded-energy low-tension maps between 2-spheres. By deriving sharp estimates for the ratio of length scales on which bubbles of opposite orientation develop, we show that we can establish a ‘quantization estimate’ which constrains the energy of the map to lie near to a discrete energy spectrum. One application is to the asymptotics of the harmonic map ﬂow; we ﬁnd uniform exponential convergence in time, in the case under consideration.
The contemplation of the atlas of an airline company always offers us
something puzzling: the trajectories of the airplanes look curved, which
goes against our basic intuition, according to which the shortest path
is a straight line. One of the reasons for this paradox is nothing but
a simple geometrical fact: on the one hand our earth is round and on
the other hand the shortest path on a sphere is an arc of great circle:
a curve whose projection on a geographical map rarely coincides with a
Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Application of the Subordination Principle to the Harmonic Mappings Convex in One Direction with Shear Construction Method
The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers M have a nontrivial amount of symmetry. By this we mean that Isom(M ) is not discrete. By the well-known theorem of Myers-Steenrod [MS], this condition is equivalent to [Isom(M ) : π1 (M )] = ∞. Also note that if any cover of M has a nondiscrete isometry group, then so does its universal cover M . Our description of such M is given in Theorem 1.2 below. The proof of this theorem uses methods from Lie theory, harmonic maps,...
The production of harmonized datasets covering all the ESPON space (31 countries) at
NUTS 2 or NUTS 3 level has been recognized as the first challenge to be solved with an
absolute priority as it is a condition of continuity with previous work realized in ESPON
2006 program. It is obvious that the new ESPON 2013 project needs immediately basic
information at this level like area, population, GDP, employment, which will be used as
reference for more sophisticated analysis where this project will produce more precise
information in their specific fields.