TAILIEUCHUNG - Báo cáo toán học: "Symmetric Laman theorems for the groups C2 and Cs"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Symmetric Laman theorems for the groups C2 and Cs. | Symmetric Laman theorems for the groups C2 and Cs Bernd Schulze Institute of Mathematics MA 6-2 TU Berlin 10623 Berlin Germany bschulze@ Submitted Jun 17 2010 Accepted Nov 3 2010 Published Nov 19 2010 Mathematics Subject Classifications 52C25 70B99 05C99 Abstract For a bar and joint framework G p with point group C3 which describes 3-fold rotational symmetry in the plane it was recently shown in Schulze Discret. Comp. Geom. 44 946-972 that the standard Laman conditions together with the condition derived in Connelly et al. Int. J. Solids Struct. 46 762-773 that no vertices are fixed by the automorphism corresponding to the 3-fold rotation geometrically no vertices are placed on the center of rotation are both necessary and sufficient for G p to be isostatic provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In this paper we prove the analogous Laman-type conjectures for the groups C2 and Cs which are generated by a half-turn and a reflection in the plane respectively. In addition analogously to the results in Schulze Discret. Comp. Geom. 44 946-972 we also characterize symmetry generic isostatic graphs for the groups C2 and Cs in terms of inductive Henneberg-type constructions as well as Crapo-type 3Tree2 partitions - the full sweep of methods used for the simpler problem without symmetry. 1 Introduction The major problem in generic rigidity is to find a combinatorial characterization of graphs whose generic realizations as bar-and-joint frameworks in d-space are rigid. While for dimension d 3 only partial results for this problem have been found it is completely solved for dimension 2. In fact using a number of both algebraic and combinatorial techniques a series of characterizations of generically rigid graphs in the plane have been established ranging from Laman s famous counts from 1970 on the number of vertices Research for this article was supported in part under a grant from NSERC .

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