Let W a be an affine Weyl group and : W a W 0 be the natural projection to the corresponding finite Weyl group. We say that w W a has finite Coxeter part if (w) is conjugate to a Coxeter element of W 0. The elements with finite Coxeter part are a union of conjugacy classes of W a. We show that for each conjugacy class O of W a with finite Coxeter part there exists a unique maximal proper parabolic subgroup W J of W a, such that the set of minimal length elements in O is exactly the set of Coxeter elements in W J. Similar results hold for twisted conjugacy classes.

In this paper we study quantum state transfer (also called quantum tunneling) on graphs when there is a potential function on the vertex set. We present two main results. First, we show that for paths of length greater than three, there is no potential on the vertices of the path for which perfect state transfer between the endpoints can occur. In particular, this answers a question raised by Godsil in Section 20 of [8]. Second, we show that if a graph has two vertices that share a common neighborhood, then there is a potential on the vertex set for which perfect state transfer will occur between those two vertices. This gives numerous examples where perfect state transfer does not occur without the potential, but adding a potential makes perfect state transfer possible. In addition, we investigate perfect state transfer on graph products, which gives further examples where perfect state transfer can occur.

Three-dimensional data merging is vital for full-field three-dimensional (3D) shape measurement. All 3D range data patches, acquired from either different sensors or the same sensor in different viewing angles, have to be merged into a single piece to facilitate future data analysis. A novel method for 3D data merging using Holoimage is proposed. Similar to the 3D shape measurement system using a phase-shifting method, Holoimage is a phase-shiftingbased computer synthesized fringe image. The 3D information is retrieved from Holoimage using a phase-shifting method. If two patches of 3D data with overlapping areas are rendered by OpenGL, the overlapping areas are resolved by the graphics pipeline, that is, only the front geometry can be visualized. Therefore, the merging is performed if the front geometry information can be obtained. Holoimage is to obtain the front geometry by projecting the fringe

Based on examples from superstring/D-brane theory since the work of Douglas and Moore on resolution of singularities of a superstring target-space Y via a D-brane probe, the richness and the complexity of the stack of punctual D0-branes on a variety, and as a guiding question, we lay down a conjecture that any resolution Y of a variety Y over Y can be factored through an embedding of Y into the stack Y of punctual D0-branes of rank Y on Y for Y in Y , where Y depends on the germ of singularities of Y . We prove that this conjecture holds for the resolution Y of a reduced singular curve Y over Y . In string-theoretical language, this says that the resolution Y of a singular curve Y always arises from an appropriate D0-brane aggregation on Y and that the rank of the Chan-Paton module of the D0-branes involved can be chosen to be arbitrarily large.

METHODS AND APPLICATIONS OF ANALYSIS. c 2009 International Press Vol. 16, No. 4, pp. 507520, December 2009 006