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• W2951686435 abstract "Let $X = left{x_1, dots, x_Nright} subset mathbb{T}^d cong [0,1]^d$ be a set of $N$ points in the $d-$dimensional torus that we want to arrange as regularly possible. The purpose of this paper is to introduce a curious energy functional $$ E(X) = sum_{1 leq m,n leq N atop m neq n} prod_{k=1}^{d}{ (1 - log{left(2 sin{ left( pi |x_{m,k} - y_{m,k} |right)} right)})}$$ and to suggest that moving a set $X$ into the direction $-nabla E(X)$ may have the effect of increasing regularity of the set in the sense of decreasing discrepancy. We numerically demonstrate the effect for Halton, Hammersley, Kronecker, Niederreiter and Sobol sets. Lattices in $d=2$ are critical points of the energy functional, some (possibly all) are strict local minima." @default.
• W2951686435 created "2019-06-27" @default.
• W2951686435 creator A5054778975 @default.
• W2951686435 date "2019-02-01" @default.
• W2951686435 modified "2023-09-27" @default.
• W2951686435 title "A Nonlocal Functional Promoting Low-Discrepancy Point Sets" @default.
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