Funktionen von beschränkter Variatiou in der Theorie der Gleichverteilung

Hlawka, Edmund

doi:10.1007/bf02415361

Cited by 198 publications

(51 citation statements)

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“…In Appendix A we prove a version of the Hlawka-Zaremba identity [32,38,39] relating the expression above to the integral of the discrepancy function, with a non-standard measure and a singularity at the origin:…”

Section: Proof Of Theorem 31mentioning

confidence: 99%

Backreaction and continuum limit in a closed universe filled with black holes

Korzyński

2014

Class. Quantum Grav.

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Abstract. We discuss the continuum limit of the initial data for a vacuum, closed cosmological model with black holes as the only sources of the gravitational field. The model we consider is an exact solution of the constraint equations and represents a vacuum universe with a number of black holes placed on a spatial slice of S 3 topology considered at the moment of its largest expansion when the black holes are momentary at rest. We explain how and under what conditions the FLRW metric arises as the continuum limit when the number of black holes contained in the model goes to infinity. We also discuss the relation between the effective cosmological parameters of the model, inferred from the large scale geometry of the spacetime, and the masses of individual black holes. In particular, we prove an estimate for the difference between the total effective mass of the system and the sum of the masses of all black holes, thus quantifying the effects of the inhomogeneities in the matter distribution or the cosmological backreaction.

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Section: Proof Of Theorem 31mentioning

confidence: 99%

Backreaction and continuum limit in a closed universe filled with black holes

Korzyński

2014

Class. Quantum Grav.

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“…Note that the case α = 1 has been studied extensively [5,7,9,12,14]. The case α > 1 has been studied recently in [2,3].…”

Section: A Koksma-hlawka Type Inequality Of Fractional Ordermentioning

confidence: 98%

“…, y s ). The inner product generalizes similarly and is given by [7,9,14] that the error using a quasi-Monte Carlo rule for integration of functions in H 1,s , or similar spaces, is small if the quadrature points are uniformly distributed. In order to quantify the distribution properties of the quadrature points certain measures have been introduced.…”

Section: Introductionmentioning

confidence: 98%

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Koksma–Hlawka type inequalities of fractional order

Dick

2007

Annali di Matematica

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The Koksma-Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded variation. Hence here we relax the smoothness assumptions required in the Koksma-Hlawka inequality. We introduce Banach spaces of functions whose fractional derivative of order α > 0 is in L p . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma-Hlawka inequality to functions whose partial fractional derivatives are in L p . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation but satisfy weaker smoothness conditions.

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“…Extending to an arbitrary number of dimensions [3] a theorem due to Koksrna [5], he showed that if /(x) (x = <x a) ..... x(O>) was of bounded variation in the sense of Hardy and Krause (for a precise definition of these concepts, see e.g. [3]), then…”

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confidence: 99%

Good lattice points in the sense of Hlawka and Monte-Carlo integration

Zaremba

1968

Monatshefte f�r Mathematik

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Funktionen von beschränkter Variatiou in der Theorie der Gleichverteilung

Cited by 198 publications

References 0 publications

Backreaction and continuum limit in a closed universe filled with black holes

Backreaction and continuum limit in a closed universe filled with black holes

Koksma–Hlawka type inequalities of fractional order

Good lattice points in the sense of Hlawka and Monte-Carlo integration

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