scholarly journals Asymptotics of $L_p$-error for adaptive approximation of $n$-variable functions by harmonic splines

2021 ◽  
Vol 19 ◽  
pp. 71
Author(s):  
T.Yu. Leskevich
Keyword(s):  
Differentiable Function ◽  
Unit Cube ◽  
Optimal Sequence ◽  
Asymptotically Optimal ◽  
Adaptive Approximation ◽  
Dimensional Unit

For a twice continuously differentiable function, defined on $n$-dimensional unit cube, we obtain sharp asymptotics of $L_p$-error for approximation by harmonic splines, and construct the asymptotically optimal sequence of partitions.

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.

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.

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

scholarly journals On Proinov’s Lower Bound for the Diaphony

2020 ◽  
Vol 15 (2) ◽  
pp. 39-72
Author(s):  
Nathan Kirk
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Lower Bounds ◽  
Unit Cube ◽  
Integral Approximation ◽  
Quantitative Theory ◽  
Dimensional Unit ◽  
Explicit Lower Bound ◽  
Irregularities Of Distribution ◽  
Infinite Sequences

AbstractIn 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2-discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.

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