Notes on the uniform distribution of sequences of integers

Kuipers, Lauwerens; Uchiyama, Saburô

doi:10.3792/pja/1195521076

Cited by 193 publications

(271 citation statements)

References 6 publications

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“…It has been shown [9] that D*N(X) > 1/2TV for any TV numbers xx, ... , xN in 7, and that the lower bound is attained for the point set …”

Section: Computational Resultsmentioning

confidence: 99%

“…An estimate of the error term e^'(r) is obtained by using the following inequality, which is due to Koksma (we refer to [9,16]). …”

Section: Error Estimatesmentioning

confidence: 99%

“…We set ¿"j = ^»',vf. 1 < i" < A^, X<j<N. For a function g on Is we denote by V(g) its variation in the sense of Hardy and Krause (we refer to [9,16] for the definition of this concept). We specialize to obtain a more useful version of Lemma 2.4.…”

Section: Error Estimatesmentioning

confidence: 99%

“…*-discrepancy, of the first P terms of the sequence. We refer to [9,16] for further information on the concept of discrepancy. We will measure the error at time r of the QMC method by the *-discrepancy of w{"', ... , vj^ relative to the exact velocity distribution at time t(n).…”

Section: Introductionmentioning

confidence: 99%

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A quasi-Monte Carlo method for the Boltzmann equation

Lécot¹

1991

Math. Comp.

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Abstract. A new quasi-Monte Carlo method for solving the Boltzmann equation in a simplified case is described. The analysis is restricted to a spatially homogeneous and isotropic gas; in addition, the molecular model only involves isotropic scattering. The scheme makes use of particles and combines an Euler scheme with numerical integrations. The sequence which is used for the quadratures must possess some symmetry properties which prescribe energy conservation for colliding particles. The error of the method is estimated by means of the discrepancy of the sequence which performs the quadratures. An algorithm for generating convenient sequences is proposed. In an example, where an exact solution is known, the computation of effective errors is included.

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“…It has been shown [9] that D*N(X) > 1/2TV for any TV numbers xx, ... , xN in 7, and that the lower bound is attained for the point set …”

Section: Computational Resultsmentioning

confidence: 99%

“…An estimate of the error term e^'(r) is obtained by using the following inequality, which is due to Koksma (we refer to [9,16]). …”

Section: Error Estimatesmentioning

confidence: 99%

Section: Error Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A quasi-Monte Carlo method for the Boltzmann equation

Lécot¹

1991

Math. Comp.

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“…and since 1 √ 2 is irrational, this implies that α is equidistributed in the interval [0, 1) (see for instance [3]). …”

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confidence: 99%

On a problem of Ahlswede and Katona

Wagner

Wang

2009

Studia Scientiarum Mathematicarum Hungarica

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Abstract. Let p(G) denote the number of pairs of adjacent edges in a graph G. Ahlswede and Katona considered the problem of maximizing p(G) over all simple graphs with a given number n of vertices and a given number N of edges. They showed that p(G) is either maximized by a quasi-complete graph or by a quasi-star. They also studied the range of N (depending on n) for which the quasi-complete graph is superior to the quasi-star (and vice versa) and formulated two questions on distributions in this context. This paper is devoted to the solution of these problems.

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Visualization approaches for the prototype improvement problem

Statnikov

Anıl

Bordetsky

et al. 2008

Multi Criteria Decision Anal

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One of the basic engineering optimization problems is improvement of the prototype. This problem is often encountered by industrial and academic organizations that produce and design various objects (e.g. motor vehicles, machine tools, ships, and aircrafts). This paper presents an approach for improving the prototype by constructing the feasible and Pareto sets while performing multicriteria analysis. We introduce visualization methods that facilitate construction of the feasible and Pareto sets. Using these techniques developed on the basis of Parameter Space Investigation method, an expert can correctly state and solve the problem under consideration in a series of dialogues with the computer. Finally, we present a case study of improving the ship prototype.

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Notes on the uniform distribution of sequences of integers

Cited by 193 publications

References 6 publications

A quasi-Monte Carlo method for the Boltzmann equation

A quasi-Monte Carlo method for the Boltzmann equation

On a problem of Ahlswede and Katona

Visualization approaches for the prototype improvement problem

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