scholarly journals Bounds for the Average $L^p$-Extreme and the $L^\infty$-Extreme Discrepancy

10.37236/1951 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Michael Gnewuch
Keyword(s):  
Upper Bounds ◽  
Unit Cube ◽  
Point Sets ◽  
Dimensional Unit ◽  
Axis Parallel ◽  
Geometric Discrepancy

The extreme or unanchored discrepancy is the geometric discrepancy of point sets in the $d$-dimensional unit cube with respect to the set system of axis-parallel boxes. For $2\leq p < \infty$ we provide upper bounds for the average $L^p$-extreme discrepancy. With these bounds we are able to derive upper bounds for the inverse of the $L^\infty$-extreme discrepancy with optimal dependence on the dimension $d$ and explicitly given constants.

scholarly journals Construction of Minimal Bracketing Covers for Rectangles

10.37236/819 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Michael Gnewuch
Keyword(s):  
Asymptotic Expansion ◽  
Lower Bound ◽  
Unit Cube ◽  
Two Dimensional ◽  
Minimal Cardinality ◽  
Significant Term ◽  
Dimensional Unit ◽  
Axis Parallel ◽  
Geometric Discrepancy

We construct explicit $\delta$-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these $\delta$-bracketing covers are bounded from above by $\delta^{-2} + o(\delta^{-2})$. A lower bound for the cardinality of arbitrary $\delta$-bracketing covers for $d$-dimensional anchored boxes from [M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24 (2008) 154-172] implies the lower bound $\delta^{-2}+O(\delta^{-1})$ in dimension $d=2$, showing that our constructed covers are (essentially) optimal. We study also other $\delta$-bracketing covers for the set system of rectangles, deduce the coefficient of the most significant term $\delta^{-2}$ in the asymptotic expansion of their cardinality, and compute their cardinality for explicit values of $\delta$.

scholarly journals Metrical Star Discrepancy Bounds for Lacunary Subsequences of Digital Kronecker-Sequences and Polynomial Tractability

2018 ◽  
Vol 13 (1) ◽  
pp. 65-86 ◽  
Author(s):  
Mario Neumüller ◽  
Friedrich Pillichshammer
Keyword(s):  
Quantitative Measure ◽  
Explicit Construction ◽  
Unit Cube ◽  
Point Sets ◽  
Finite Point ◽  
Quasi Monte Carlo ◽  
Monte Carlo Algorithms ◽  
Dimensional Unit ◽  
Point Set ◽  
Star Discrepancy

Abstract The star discrepancy $D_N^* \left( {\cal P} \right)$ is a quantitative measure for the irregularity of distribution of a finite point set 𝒫 in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets 𝒫 in [0, 1)d with |𝒫| = N and $D_N^* \left( {\cal P} \right) = O\left( {\left( {\log \,N} \right)^{d - 1} /N} \right)$ . However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g., for N ≤ ed−1). In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2there exist point sets 𝒫 in [0, 1)d with |𝒫| = N and $D_N^* \left( {\cal P} \right) \le C\sqrt {d/N}$ . Although not optimal in an asymptotic sense in N, this upper bound has a much better (and even optimal) dependence on the dimension d. Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (nα)-sequence and showed a metrical discrepancy bound of the form $C\sqrt {d\left({\log \,d} \right)/N}$ with implied absolute constant C> 0 independent of N and d. In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

A travelling salesman problem in the k-dimensional unit cube

1992 ◽  
Vol 11 (1) ◽  
pp. 19-21 ◽  
Author(s):  
B. Bollobás ◽  
A. Meir
Keyword(s):  
Travelling Salesman Problem ◽  
Unit Cube ◽  
Travelling Salesman ◽  
Dimensional Unit

scholarly journals The number of lattice rules having given invariants

1992 ◽  
Vol 46 (3) ◽  
pp. 479-495 ◽  
Author(s):  
Stephen Joe ◽  
David C. Hunt
Keyword(s):  
Normal Form ◽  
Quadrature Rule ◽  
Unit Cube ◽  
Numerical Results ◽  
Smith Normal Form ◽  
Lattice Rules ◽  
Dimensional Unit ◽  
Lattice Rule ◽  
Positive Integers ◽  
Theoretical Results

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

Constraint satisfaction problems on specific subsets of the n-dimensional unit cube

2015 ◽  
Author(s):  
Levon Aslanyan ◽  
Hasmik Sahakyan ◽  
Hans-Dietrich Gronau ◽  
Peter Wagner
Keyword(s):  
Constraint Satisfaction ◽  
Unit Cube ◽  
Constraint Satisfaction Problems ◽  
Dimensional Unit

Hamiltonian Circuits and Paths on the n-Cube

1966 ◽  
Vol 9 (05) ◽  
pp. 557-562 ◽  
Author(s):  
H. L. Abbott
Keyword(s):  
Unit Cube ◽  
Dimensional Unit ◽  
Hamiltonian Circuits

For positive integral n let Cn denote the n-dimensional unit cube with vertices (δ1, δ2,…, δn) where δi = 0 or 1 for i=1, 2,…, n. Call two vertices of Cn adjacent if the distance between them is 1.

Mean and variance of vacancy for distribution of k-dimensional spheres within k-dimensional space

1984 ◽  
Vol 21 (4) ◽  
pp. 738-752 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Dimensional Space ◽  
Sufficient Conditions ◽  
Unit Cube ◽  
Necessary And Sufficient Conditions ◽  
Dimensional Sphere ◽  
Asymptotic Formulae ◽  
Dimensional Unit ◽  
Mean And Variance ◽  
The Mean ◽  
Necessary And Sufficient

Let n points be distributed independently within a k-dimensional unit cube according to density f. At each point, construct a k-dimensional sphere of content an. Let V denote the vacancy, or ‘volume' not covered by the spheres. We derive asymptotic formulae for the mean and variance of V, as n → ∞and an → 0. The formulae separate naturally into three cases, corresponding to nan → 0, nan → a (0 < a <∞) and nan →∞, respectively. We apply the formulae to derive necessary and sufficient conditions for V/E(V) → 1 in L2.

scholarly journals Covering the k-skeleton of the 3-dimensional unit cube by six balls

2014 ◽  
Vol 336 ◽  
pp. 85-95 ◽  
Author(s):  
A. Joós
Keyword(s):  
Unit Cube ◽  
3 Dimensional ◽  
Dimensional Unit

Covering the 3-dimensional unit cube by six rectangular boxes

2014 ◽  
Vol 56 (1) ◽  
pp. 63-74 ◽  
Author(s):  
Gábor Horváth ◽  
István Talata
Keyword(s):  
Unit Cube ◽  
3 Dimensional ◽  
Dimensional Unit
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