Symmetry And Group

Relation of Preferential Motion and of the Spectral-Class by Perrine C. D.

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By Perrine C. D.

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4), so it remains to check that if A is a seam of Q, then (A,j)Vr ml = ( A , j ) Z p~ for r = 1,2 and all j (mod M). Fig. 5. Flows around a vertex (type 1) Now, consider a vertex of Q of type r (r = 1, 2) and collapsing index i that contains the seams A = A1, A 2 , . . , Ak in counterclockwise order, as shown in Fig. 5. -. + tk be the total flow into the vertex. 5) = ( A k , j + t l + . . + tk-1)Vr = ( A I , j + t). However, since the definition of collapsing index (Defn. 1) implies that ki = mr, and the definition of quilt diagram (Defn.

2 . 2 . A seam, m2 -- 2 We begin by describing the basic building block of our 2-complexes. Consider the double triangular region shown in Fig. 1. Such a region is called a seam. More formally, a seam is a 2-complex s constructed as follows. 1. s has four 0-cells, which are marked in Fig. 1 as dl and d2 (the dots of type 1 and 2), pt, and Pr. By convention, black dots are of type 1, and white dots are of type 2. 2. s has five 1-cells, which consist of one solid 1-cell, between dl and d2 (the midline of s), and four dotted 1-cells, between dl and Pt, dl and Pr, d2 and Pt, and dz and Pr (marked V1, Vx-1, V2-1, and V2, respectively).

We first define the case n -- c~ (Defn. 1), and later proceed to the general case (Defn. 6). 1. A (ml,m2,c~)-modular quilt Q is a set of seams with some of their dotted 1-cells identified (in the sense of a quotient space) according to the following rules. 1. For r = 1, 2, a dotted 1-cell marked Vr can only be identified with a dotted 1-cell marked V~-1. Furthermore, all identifications are cellular (n-cells with n-cells), and dots of type r (r = 1, 2) m a y only be identified with other dots of type r.

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