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The discussion may indeed be concerned with homogeneous Relations 22 relations; it may, however, be principally about heterogeneous relations holding between two sets that are identical just by coincidence. This leads to two different forms of graph representation, as in Fig. 8. visualized as a 1-graph a a b b c c d d a b c d a d b a b c d 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 visualized as a bipartitioned graph c Fig. 8 Two different types of graph representing the same relation Just looking at the relation and without knowing whether the homogeneous or the heterogeneous form was intended, one cannot decide readily on one or the other graph presentation.

They may be given as a matrix, a sequence, via cycles, or as a function. We provide mechanisms to convert between these forms and to apply permutations to some set. 1 1⎛ 0 2 0 3⎜ 0 4⎜ 0 5⎝ 0 6 0 7 1 2 0 0 0 0 0 1 0 3 0 1 0 0 0 0 0 4 1 0 0 0 0 0 0 5 0 0 1 0 0 0 0 6 0 0 0 0 1 0 0 7 0⎞ 0 0⎟ 1⎟ 0⎠ 0 0 1 1⎛ 0 4 0 7⎜ 1 2⎜ 0 3⎝ 0 5 0 6 0 4 1 0 0 0 0 0 0 7 0 1 0 0 0 0 0 2 0 0 0 0 0 0 1 3 0 0 0 1 0 0 0 5 0 0 0 0 1 0 0 6 0⎞ 0 0⎟ 0⎟ 0⎠ 1 0 Fig. 17 A permutation rearranged to cycle form One will easily confirm that both matrices in Fig.

Colors,weekdays = red green blue orange 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 Mon Tue Wed Thu Fri Sat There is also a tradition of denoting this differently,2 which we do not follow. It makes no sense to unite, for example, relations of different types, and therefore this is not allowed. Concerning the top and bottom relations, we have been a bit sloppy here. The symbols for the empty or null and for the universal relation should have been X,Y and X,Y , respectively. While we know the typing in cases of union, intersection, and negation from the operands R, S, we should provide this information explicitly for the null and the universal relation.

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Mountain pass solutions for a mean field equation from two-dimensional turbulence by Ricciardi T.


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