The results obtained by modeling an infinite linear array of electrically short dipoles by an array of pseudopotentials (the pseudopotential is a quantum-mechanical analog of the dipole) can be used for the analysis of finite linear dipole arrays. This idea is utilized to investigate properties of finite arrays such as resonances, grating lobes, and end effects. For a wide range of parameter values, it is found that the pseudopotentials can quantitatively describe all aforementioned properties of the actual finite array. Certain extensions to waveguide arrays are discussed.
The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on highdimensional spaces and present a new randomized algorithm which efficiently computes approximate rnets with respect to Euclidean distance. For any fixed > 0, the approximation factor is 1 + and the complexity is polynomial in the dimension and subquadratic in the number of points. The algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSHbased) construction of Eppstein et al. [EHS15] in terms of complexity, by reducing the dependence on , provided that is sufficiently small. Our method does not require LSH but, instead, follows Valiant's [Val15] approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the (1 + )-approximate kth nearest neighbor distance in time subquadratic in the size of the input.
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