Applications of graph theory to group structure by Claude Flament

By Claude Flament

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In particular, if f α is consistently the identity, then ν(z α , z0α ) = ν(z 0 , z00 ). The assumption that f be C 1 is made only to ensure that w = f ◦ z is piecewise differentiable. Hence, we can relax this assumption, and require differentiability of f only in a neighborhood of z(U ). 18/31), provided that 0 ∈ z(U ). However, for the theory to be derived now, we have to extend the definition of the induced winding number to the case where z0 = 0 and the curve z passes through the origin. Here, the integral defining ν(w, w0 ) does not necessarily exist, but we can resort to the following smoothing process: Let f α be a continuous family of C 1 -functions converging uniformly according to lim f α − f α→0 ∞ = 0.

For a regular C 2 -surface x, let Dx+ := (I −1 Dx)t denote the pseudo-inverse of Dx, which is a (3 × 2)-matrix. Then we define the embedded Weingarten map of x as the symmetric (3 × 3)-matrix W := Dx+ II · Dx+ . We claim that this object is a geometric invariant that contains the complete curvature information in a continuous way. Because W · n = 0, the normal vector is always an eigenvector of W to the eigenvalue 0. 2 Regular Surfaces 21 (2 × 3)-matrix R. The diagonal matrix of the corresponding pair of eigenvalues is denoted by K := diag(κ1 , κ2 ).

An equally valid argument would be to employ a rigid embedding ϕ of abutting cells. The resulting surface x◦ϕ−1 has a well defined normal vector. 2/18, the patches equally have a well defined normal vector which satisfies the consistency condition. 5/22 that for a regular C 2 -spline surface x the embedded Weingarten map W:S (σ, i) → Wi (σ) ∈ R3×3 is well defined as a spline in C 0 (S, R3×3 ), where Wi denotes the embedded Weingarten map of the patch xi . 4 Crk-Splines As stated earlier, C 1 -continuity of splines is not necessary for geometric smoothness, but is assumed for convenience.

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