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Register a new userEnumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes
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For a positive integer $d$, the $d$-dimensional Chebyshev-Frolov lattice is the $mathbb{Z}$-lattice in $mathbb{R}^d$ generated by the Vandermonde matrix associated to the roots of the $d$-dimensional Chebyshev polynomial. It is important to enumerate the points from the Chebyshev-Frolov lattices in axis-parallel boxes when $d = 2^n$ for a non-negative integer $n$, since the points are used for the nodes of Frolovs cubature formula, which achieves the optimal rate of convergence for many spaces of functions with bounded mixed derivatives and compact support. The existing enumeration algorithm for such points by Kacwin, Oettershagen and Ullrich is efficient up to dimension $d=16$. In this paper we suggest a new enumeration algorithm of such points for $d=2^n$, efficient up to $d=32$.
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We show that, for every set of $n$ points in the $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $Omega(d/n)$ as $ntoinfty$ and $d$ is fixed. In the opposite direction, we give a construction without an empty axis-parallel box of volume $O(d^2log d/n)$. These improve on the previous best bounds of $Omega(log d/n)$ and $O(2^{7d}/n)$ respectively.
It is proved that the Fibonacci and the Frolov point sets, which are known to be very good for numerical integration, have optimal rate of decay of dispersion with respect to the cardinality of sets. This implies that the Fibonacci and the Frolov point sets provide universal discretization of the uniform norm for natural collections of subspaces of the multivariate trigonometric polynomials. It is shown how the optimal upper bounds for dispersion can be derived from the upper bounds for a new characteristic -- the smooth fixed volume discrepancy. It is proved that the Fibonacci point sets provide the universal discretization of all integral norms.
Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called unified method in the interior of a convex polygon provides an example of the applicability of these analytic expressions.
In this paper, we consider various classes of polyiamonds that are animals residing on the triangular lattice. By careful analyses through certain layer-by-layer decompositions and cell pruning/growing arguments, we derive explicit forms for the generating functions of the number of nonempty translation-invariant baryiamonds (bargraphs in the triangular lattice), column-convex polyiamonds, and convex polyiamonds with respect to their perimeter. In particular, we show that the number of (A) baryiamonds of perimeter $n$ is asymptotically $$frac{(xi+1)^2sqrt{xi^4+xi^3-2xi+1}}{2sqrt{pi n^3}}xi^{-n-2},$$ where $xi$ is a root of a certain explicit polynomial of degree 5. (B) column-convex polyiamonds of perimeter $n$ is asymptotic to $$frac{(17997809sqrt{17}+3^3cdot13cdot175463)sqrt{95sqrt{17}-119}}{2^7cdot43^2cdot 89^2sqrt{6pi n^3}}left(frac{3+sqrt{17}}{2}right)^{n-1}.$$ (C) convex polyiamonds of perimeter $n$ is asymptotic to $$frac{1280}{441sqrt{3pi n^3}}3^n.$$
We describe a new parallel implementation, mplrs, of the vertex enumeration code lrs that uses the MPI parallel environment and can be run on a network of computers. The implementation makes use of a C wrapper that essentially uses the existing lrs code with only minor modifications. mplrs was derived from the earlier parallel implementation plrs, written by G. Roumanis in C++. plrs uses the Boost library and runs on a shared memory machine. In developing mplrs we discovered a method of balancing the parallel tree search, called budgeting, that greatly improves parallelization beyond the bottleneck encountered previously at around 32 cores. This method can be readily adapted for use in other reverse search enumeration codes. We also report some preliminary computational results comparing parallel and sequential codes for vertex/facet enumeration problems for convex polyhedra. The problems chosen span the range from simple to highly degenerate polytopes. For most problems tested, the results clearly show the advantage of using the parallel implementation mplrs of the reverse search based code lrs, even when as few as 8 cores are available. For some problems almost linear speedup was observed up to 1200 cores, the largest number of cores tested.
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