Modeling polarization for Hyper-NA lithography tools and masks
Kafai Lai, Alan E. Rosenbluth, et al.
SPIE Advanced Lithography 2007
We consider the following long-range percolation model: an undirected graph with the node set {0, 1, ..., N}d, has edges (x, y) selected with probability ≈ β/||x - y||s if ||x - y|| > 1, and with probability 1 if ||x - y|| = 1, for some parameters β, s > 0. This model was introduced by Benjamini and Berger, who obtained bounds on the diameter of this graph for the one-dimensional case d = 1 and for various values of s, but left cases s = 1, 2 open. We show that, with high probability, the diameter of this graph is Θ(log N/log log N) when s = d, and, for some constants 0 < η1 < η2 < 1, it is at most Nη2 when s = 2d, and is at least Nη1 when d = 1, s = 2, β < 1 or when s > 2d. We also provide a simple proof that the diameter is at most logO(1) N with high probability, when d < s < 2d, established previously in [2]. © 2002 Wiley Periodicals, Inc.
Kafai Lai, Alan E. Rosenbluth, et al.
SPIE Advanced Lithography 2007
David Gamarnik
IEEE TACON
Tong Zhang, G.H. Golub, et al.
Linear Algebra and Its Applications
Salvatore Certo, Anh Pham, et al.
Quantum Machine Intelligence