We study equations over torsion-free groups in terms of their “t–shape” (the occurences of the variable t in the equation). A t–shape is good if any equation with that shape has a solution. It is an outstanding conjecture [5] that all t–shapes are good. In [2] we proved the conjecture for a large class of t–shapes called amenable. In [1] Clifford and Goldstein characterised a class of good t–sh...

Curvy roads are important subjects for many applications, because of their spatial geometry. Digital curve identification is easier from continuous objects than from discrete objects, such as roads extracted from remote sensing data. Use of high spatial resolution images can lead to an increase in the accuracy of the detection , and it also gives the possibility to detect curvy segments from th...

In recent years, more 3D protein structures have become available, which has made the analysis of large molecular structures much easier. There is a strong demand for geometric models for the study of protein-related interactions. Alpha shape and Delaunay triangulation are powerful tools to represent protein structures and have advantages in characterizing the surface curvature and atom contact...

Despite considerable efforts, description of molecular shape is still largely an unresolved problem. Given the importance of molecular shape in the description of spatial interactions in crystals or ligand-target complexes, this is not a satisfying state. In the current work, we propose a novel application of alpha shapes to the description of the shapes of small molecules. Alpha shapes are par...

Morphological filters, evolved from the traditional envelope filter, are function oriented filtration techniques. A recent research on the implementation of morphological filters was based on the theoretical link between morphological operations and the alpha shape. However the Delaunay triangulation on which the alpha shape method depends is costly for large areal data. This paper proposes a d...

In this paper, we present a new set of blending operations for implicitly defined geometric shapes. The proposed shape operators are piecewise polynomial and blending range controllable, and can be constructed to any required degree of smoothness. The key idea behind these techniques is the introduction of the concept of the smooth absolute functions, which in turn lead to the definition of smo...