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By Ben Ayed M., El Mehdi K., Grossi M.

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Extra resources for Asymptotic behavior of least energy solutions of a biharmonic equation in dimension four

Sample text

Regularity for lines is defined dually. A point x is coregular provided each line incident with x is regular. 3 of Payne and Thas [128]. • All lines of the GQ Q(d, q) are regular, with d e {3,4,5}. • All points of Q(3, q) are regular, for q even all points of Q(4,q) are regular, for q odd no point of Q(4, g) is regular, and for any prime power q no point of Q(5, q) is regular. • No point and no line of H(4, q2) is regular. • The point (oo) of the GQ T 2 (C), respectively T 3 (C), is coregular. If T 2 (0) has at least one regular pair of nonconcurrent lines of Type li Chapter 2.

I) Each point is incident with exactly one element of B\, each line and each circle are incident with exactly one common point. (ii) Any three distinct points, no two of which are incident with a common line, are incident with exactly one common circle. (iii) If x and y are distinct noncollinear points of £ and if C is a circle incident with x but not with y, then there is just one circle C incident with x, y and having only x in common with C (two circles having exactly one point x in common will be called tangent circles at x).

From the proof of (i) easily follows that any triad {x, y, z} has either 0 or 2 centers. Conversely, if \{x,y, z}x\ e {0,2} for any triad {x, y, z}, then it is clear that the pair {x, y} is antiregular. (iii). Assume s = t ^ 1, and that the pair of noncollinear points {x,y} is antiregular. Let L be a line not incident with x or y, and not incident with any point of {x, y}-1. , us}. The notation is chosen in such a way that xv0Iuo and yvilui. , s, has 0 or 2 centers. , vs}, it easily follows that s - 1 is even.