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A simple short proof of the Johnson-Lindenstrauss lemma (concerning nearly isometric embeddings of finite point sets in lower-dimensional spaces) is given. This result is applied to show that if G is a graph on n vertices and with smallest eigen-value i then its sphericity sph(G) is less than cA2 log n. It is also proved that if G or its complement is a… (More)
We prove that if the three angles of a triangle T in the plane are different from (60~176176 (30 ~ 30 ~ 120~ (45~176176 ~ 60~176 then the set of vertices of those triangles which are obtained from T by repeating 'edge-reflection' is everywhere dense in the plane.
Consider a unit sphere on which are placed N random spherical caps of area 4rip(N). We prove that if lim (p(N).N/log N) < 1, then the probability that the sphere is completely covered by N caps tends to 0 as N-" 0% and if lim (p(N).N/ log N) > 1, then for any integer n > 0 the probability that each point of the sphere is covered more than n times tends to 1… (More)