By Neil White
This quantity, the 3rd in a chain that started with the speculation of Matroids (1986) and Combinatorial Geometries (1987), concentrates at the purposes of matroid conception to quite a few themes from geometry (rigidity and lattices), combinatorics (graphs, codes, and designs) and operations learn (the grasping algorithm).
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Extra resources for Matroid applications
Assume that generic rigidity holds for all triangulated spheres with 4 vertices. We first show that there is a `shrinkable edge'. Any non-shrinkable edge e is part of a non-facial triangle separating a disc of N interior vertices from the remaining triangulated sphere. We claim that there is a shrinkable edge inside each such disc. If N = 1, this vertex is 3-valent, and every interior edge is shrinkable. Assume a shrinkable edge occurs for N = k.
Vertex splitting theorem for conic-rigidity. , (1, k + m) with p1- P2 not parallel to p1- p3, then for any k + m > 2, the new geometric graph on the vertex split on (1, 2), (1, 3) is conic-independent for almost all positions for the new vertex po. Proof. 6, choosing, as a limiting initial case, to add po at p1, with the `line' po, p1 assigned a direction D01 so that 2 2 2 2 = D12, D03 = D13, and Dot are independent 3-vectors. This creates the conic-rigidity matrix Do2 2 Dot 2 D 02 Dot 0 Do3 0 0 D12 2 2 2 0 0 -Do2 0 0 0 0 - D03 2 -D12 0 0 z 2 0 2 0 D13 0 -D13 0 0 D14 0 0 -D14 0 ...
2) Analysis of the matroid is based on the dual concepts of row dependences (self-stresses) and column dependences (infinitesimal motions). (3) If variables are used for these entries, we find the `generic' matroid for the class of geometric realizations. Independent sets, and bases, in the generic matroid can often, but not always, be characterized by simple counts (semimodular functions) on the underlying combinatorial structures. ) point to a fundamental problem in defining matroids by semimodular functions that are negative on small sets.
Matroid applications by Neil White