Formally, a rotation system is defined as a pair (σ,θ) where σ and θ are permutations acting on the same ground set B, θ is a fixed-point-free involution, and the group <σ,θ> generated by σ and θ acts transitively on B.
To derive a rotation system from a 2-cell embedding of a connected multigraph G on an oriented surface, let B consist of the darts (or flags, or half-edges) of G; that is, for each edge of G we form two elements of B, one for each endpoint of the edge. Even when an edge has the same vertex as both of its endpoints, we create two darts for that edge. We let θ(b) be the other dart formed from the same edge as b; this is clearly an involution with no fixed points. We let σ(b) be the dart in the clockwise position from b in the cyclic order of edges incident to the same vertex, where "clockwise" is defined by the orientation of the surface.
If a multigraph is embedded on an orientable but not oriented surface, it generally corresponds to two rotation systems, one for each of the two orientations of the surface. These two rotation systems have the same involution θ, but the permutation σ for one rotation system is the inverse of the corresponding permutation for the other rotation system.
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Other articles related to "formal definition":
... The above controller for crosswalk lights can be modeled by an atomic SP-DEVS model ... Formally, an atomic SP-DEVS is a 7-tuple where is a finite set of input events is a finite set of output events is a finite set of states is the initial state is the time advanced function which defines the lifespan of a state where is the set of non-negative rational numbers plus infinity ...
... In typical usage, the formal definition of O notation is not used directly rather, the O notation for a function f(x) is derived by the following simplification rules If f(x) is ... One may confirm this calculation using the formal definition let f(x) = 6x4 − 2x3 + 5 and g(x) = x4 ... Applying the formal definition from above, the statement that f(x) = O(x4) is equivalent to its expansion, for some suitable choice of x0 and M and for all x > x0 ...
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—Samuel Taylor Coleridge (17721834)
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—Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)