We introduce a concept of Simplicity which in a sense corresponds to the reverse direction to the concept of Complexity. Many problems of Asymptotic Geometric Analysis rotate around this notion. We also describe the concept of Concentration and suggest a new direction of possible applications of the Concentration Phenomenon.

The present article applies the method of Geometric Analysis to the study H-type groups satisfying the J condition and finishes the series of works describing the Heisenberg group and the quaternion H-type group. The latter class of H-type groups satisfying the J condition is related to the octonions. The relations between the group structure and the boundary of the corresponding Siegel upper h...

Given a weighted graph G = (V,E) and a real number t ≥ 1, a t-spanner of G is a spanning subgraph G′ with the property that for every edge xy in G, there exists a path between x and y in G′ whose weight is no more than t times the weight of the edge xy. We review results and present open problems on different variants of the problem of constructing plane geometric t-spanners.

Article history: Received 13 October 2010 Received in revised form 18 March 2012 Accepted 19 March 2012 Available online xxxx

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We introduce the concept of homotopic spanners in the plane with obstacles and show lower bounds on the number of edges that they require. We also provide a construction based on Θ-graphs for constructing homotopic spanners.

Constraints can be used to specify and maintain spatial relationships among objects in a geometric design. In the 3-D geometric design domain , the diversity of possible relationships among objects makes it diicult for the designer to specify useful or intended relationships in a productive and intuitive manner. We have built a constraint-based 3D geometric editor called DEVI that infers possib...