Wavefront and caustic surfaces of refractive laser beam shaper
David L. Shealy, John A. Hoffnagle
SPIE Optical Engineering + Applications 2007
The 1935 result of Erdos and Szekeres that any sequence of ≥ n 2 + 1 real numbers contains a monotonic subsequence of ≥ n + 1 terms has stimulated extensive further research, including a paper of J. B. Kruskal that defined an extension of monotonicity for higher dimensions. This paper provides a proof of a weakened form of Kruskal's conjecture for 2-dimensional Euclidean space by showing that there exist sequences of n points in the plane for which the longest monotonic subsequences have length ≤ n1/2 + 3. Weaker results are obtained for higher dimensions. When points are selected at random from reasonable distributions, the average length of the longest monotonic subsequence is shown to be ∼2n1/2 as n → ∞ for each dimension.
David L. Shealy, John A. Hoffnagle
SPIE Optical Engineering + Applications 2007
A.R. Gourlay, G. Kaye, et al.
Proceedings of SPIE 1989
Michael E. Henderson
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
William Hinsberg, Joy Cheng, et al.
SPIE Advanced Lithography 2010