A New Boundary of the Mapping Class Group

Based on the action of the mapping class group on the space of measured foliations,we construct a new boundary of the mapping class group and study the structure of this boundary.As an application,for any point in Teichmüller space,we consider the orbit of this point under the action of the mapping class group and describe the closure of this orbit in the Thurston compactification and the Gardiner-Masur compactification of Teichmüller space.We also construct some new points in the Gardiner-Masur boundary of Teichmüller space.

Quantitative volumetric analysis of the optic radiation in the normal human brain using diffusion tensor magnetic resonance imaging-based tractography

To attain the volumetric information of the optic radiation in normal human brains, we per- formed diffusion tensor imaging examination in 13 healthy volunteers. Simultaneously, we used a brain normalization method to reduce individual brain variation and increase the accuracy of volumetric information analysis. In addition, tractography-based group mapping method was also used to investigate the probability and distribution of the optic radiation pathways. Our results showed that the measured optic radiation fiber tract volume was a range of about 0.16% and that the fractional anisotropy value was about 0.53. Moreover, the optic radiation probability fiber pathway that was determined with diffusion tensor tractography-based group mapping was able to detect the location relatively accurately. We believe that our methods and results are help- ful in the study of optic radiation fiber tract information.

Equivariant Mapping Class Group and Orbit Braid Group

Motivated by the work of Birman about the relationship between mapping class groups and braid groups,the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper.Their construction is based on the exact sequence given by the fibration F_(0)^(G) M→F(M/G,n).The conclusion is closely connected with the braid group of the quotient space.Comparing with the situation without the group action,there is a big difference when the quotient space is T^(2).

Weak Graph Map Homotopy and Its Applications

The authors introduce a notion of a weak graph map homotopy(they call it M-homotopy),discuss its properties and applications.They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001.The difference between them is that the weak graph map homotopy transformation is defined in terms of maps,while the graph homotopy transformation is defined by means of combinatorial operations.They discuss its advantages over the graph homotopy transformation.As its applications,they investigate the mapping class group of a graph and the 1-order M P-homotopy group of a pointed simple graph.Moreover,they show that the 1-order M P-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.