Calculation of improper integrals using uniformly distributed sequences

Baxa, Christoph

doi:10.4064/aa119-4-5

Cited by 6 publications

(1 citation statement)

References 11 publications

(16 reference statements)

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“…]. An even stronger version was established in [Bax05], under broader, but very technical assumptions.…”

Section: Equidistribution Argumentsmentioning

confidence: 99%

Torsion homology growth beyond asymptotics

Braunling

2017

Preprint

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We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's theorem on periodic homology. Our results generalize to other rank 1 growth phenomena, e.g. Reidemeister-Franz torsion growth for higher-dimensional knots. We also analyze the exceptional cases where the meromorphic continuation does not exist.

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“…]. An even stronger version was established in [Bax05], under broader, but very technical assumptions.…”

Section: Equidistribution Argumentsmentioning

confidence: 99%

Torsion homology growth beyond asymptotics

Braunling

2017

Preprint

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Quasi-Monte Carlo algorithms for unbounded, weighted integration problems

Hartinger¹,

Kainhofer²,

Tichy³

2004

Journal of Complexity

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In this article we investigate quasi-Monte Carlo (QMC) methods for multidimensional improper integrals with respect to a measure other than the uniform distribution. Additionally, the integrand is allowed to be unbounded at the lower boundary of the integration domain. We establish convergence of the QMC estimator to the value of the improper integral under conditions involving both the integrand and the sequence used. Furthermore, we suggest a modification of an approach proposed by Hlawka and Mu¨ck for the creation of low-discrepancy sequences with regard to a given density, which are suited for singular integrands. r This paper is devoted to quasi-Monte Carlo (QMC) techniques for weighted integration problems of the formwhere H denotes an s-dimensional probability distribution with support K ¼ ½a; b ¼ Q s i¼1 ½a ðiÞ ; b ðiÞ DR s and continuous density hðxÞ: Furthermore, f is a function with singularities on the left boundary of K:Numerical integration problems of this form frequently occur in practice, e.g. in the field of computational finance. A typical example is the estimation of the mean of ARTICLE IN PRESS

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Averaging Almost Periodic Functions along Exponential Sequences

Baake

Haynes

Lenz³

Aperiodic Order

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Abstract. The goal of this expository article is a fairly self-contained account of some averaging processes of functions along sequences of the form (α n x) n∈N , where α is a fixed real number with |α| > 1 and x ∈ R is arbitrary. Such sequences appear in a multitude of situations including the spectral theory of inflation systems in aperiodic order. Due to the connection with uniform distribution theory, the results will mostly be metric in nature, which means that they hold for Lebesgue-almost every x ∈ R.

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Calculation of improper integrals using uniformly distributed sequences

Cited by 6 publications

References 11 publications

Torsion homology growth beyond asymptotics

Torsion homology growth beyond asymptotics

Quasi-Monte Carlo algorithms for unbounded, weighted integration problems

Averaging Almost Periodic Functions along Exponential Sequences

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