Low-discrepancy point sets for non-uniform measures

Christoph Aistleitner; Josef Dick

Displaying similar documents to “Low-discrepancy point sets for non-uniform measures”

Explicit construction of normal lattice configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Colloquium Mathematicae

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We extend Champernowne’s construction of normal numbers to base b to the $ℤ^{d}$ case and obtain an explicit construction of a generic point of the $ℤ^{d}$ shift transformation of the set ${0, 1, . . ., b - 1}^{ℤ^{d}}$ .

A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

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Let T be a linear operator on a Banach space X with $s u p ₙ | | T ⁿ / n^{w} | | < \infty$ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) $n^{- 1} \sum_{k = 0}^{n - 1} T^{k}$ converges uniformly; (ii) $c l (I - T) X = z \in X : l i m_{n} \sum_{k = 1}^{n} T^{k} z / k e x i s t s$ .

Lower bounds for the largest eigenvalue of the gcd matrix on ${1, 2, \dots, n}$

Jorma K. Merikoski (2016)

Czechoslovak Mathematical Journal

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Consider the $n \times n$ matrix with $(i, j)$ ’th entry $gcd (i, j)$ . Its largest eigenvalue $λ_{n}$ and sum of entries $s_{n}$ satisfy $λ_{n} > s_{n} / n$ . Because $s_{n}$ cannot be expressed algebraically as a function of $n$ , we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that $λ_{n} > 6 π^{- 2} n log n$ for all $n$ . If $n$ is large enough, this follows from F. Balatoni (1969).

Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure

Zbigniew Lipecki (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $𝔐$ and $ℜ$ be algebras of subsets of a set $Ω$ with $𝔐 \subset ℜ$ , and denote by $E (μ)$ the set of all quasi-measure extensions of a given quasi-measure $μ$ on $𝔐$ to $ℜ$ . We give some criteria for order boundedness of $E (μ)$ in $b a (ℜ)$ , in the general case as well as for atomic $μ$ . Order boundedness implies weak compactness of $E (μ)$ . We show that the converse implication holds under some assumptions on $𝔐$ , $ℜ$ and $μ$ or $μ$ alone, but not in general.

On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals

Abraham Racca, Emmanuel Cabral (2016)

Mathematica Bohemica

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Equiintegrability in a compact interval $E$ may be defined as a uniform integrability property that involves both the integrand $f_{n}$ and the corresponding primitive $F_{n}$ . The pointwise convergence of the integrands $f_{n}$ to some $f$ and the equiintegrability of the functions $f_{n}$ together imply that $f$ is also integrable with primitive $F$ and that the primitives $F_{n}$ converge uniformly to $F$ . In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...

Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures

Eric A. Carlen, Dario Cordero-Erausquin, Elliott H. Lieb (2013)

Annales de l'I.H.P. Probabilités et statistiques

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An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^{2}$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site...

Nearstandardness on a finite set

Lyantse V.

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AbstractLet T be a finite set for which card T is a natural nonstandard number. The linear space $ℂ^{T}$ of complex-valued functions on T is nonstandard. For the analysis on $ℂ^{T}$ we need a concept of nearstandardness in this space. A version how to introduce such a concept is proposed. Some elementary examples are given. CONTENTSIntroduction.................................................................................................................50. Preliminary notes....................................................................................................7 0.1....

On linear extension for interpolating sequences

Eric Amar (2008)

Studia Mathematica

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Let A be a uniform algebra on X and σ a probability measure on X. We define the Hardy spaces $H^{p} (σ)$ and the $H^{p} (σ)$ interpolating sequences S in the p-spectrum $ℳ_{p}$ of σ. We prove, under some structural hypotheses on A and σ, that if S is a “dual bounded” Carleson sequence, then S is $H^{s} (σ)$ -interpolating with a linear extension operator for s < p, provided that either p = ∞ or p ≤ 2. In the case of the unit ball of ℂⁿ we find, for instance, that if S is dual bounded in $H^{\infty} ()$ then S is $H^{p} ()$ -interpolating with...

Existence and upper semicontinuity of uniform attractors in $H ¹ (ℝ^{N})$ for nonautonomous nonclassical diffusion equations

Cung The Anh, Nguyen Duong Toan (2014)

Annales Polonici Mathematici

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We prove the existence of uniform attractors $_{ε}$ in the space $H ¹ (ℝ^{N})$ for the nonautonomous nonclassical diffusion equation $u_{t} - ε Δ u_{t} - Δ u + f (x, u) + λ u = g (x, t)$ , ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ${_{ε}}_{ε \in [0, 1]}$ at ε = 0 is also studied.

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

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Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $_{ν} (X)$ and $_{ν} (X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

On inhomogeneous self-similar measures and their $L^{q}$ spectra

Przemysław Liszka (2013)

Annales Polonici Mathematici

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Let $S_{i} : ℝ^{d} \to ℝ^{d}$ for i = 1,..., N be contracting similarities, let $(p ₁, . . ., p_{N}, p)$ be a probability vector and let ν be a probability measure on $ℝ^{d}$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on $ℝ^{d}$ such that $μ = \sum_{i = 1}^{N} p_{i} μ \circ S_{i}^{- 1} + p ν$ . We give satisfactory estimates for the lower and upper bounds of the $L^{q}$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...

On area and side lengths of triangles in normed planes

Gennadiy Averkov, Horst Martini (2009)

Colloquium Mathematicae

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Let $ℳ^{d}$ be a d-dimensional normed space with norm ||·|| and let B be the unit ball in $ℳ^{d}$ . Let us fix a Lebesgue measure $V_{B}$ in $ℳ^{d}$ with $V_{B} (B) = 1$ . This measure will play the role of the volume in $ℳ^{d}$ . We consider an arbitrary simplex T in $ℳ^{d}$ with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of $V_{B} (T)$ are determined. For d ≥ 3 it is noticed that the tight lower bound of $V_{B} (T)$ is zero.

A tight quantitative version of Arrow’s impossibility theorem

Nathan Keller (2012)

Journal of the European Mathematical Society

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The well-known Impossibility Theorem of Arrow asserts that any generalized social welfare function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any $ϵ > 0$ , there exists $δ = δ (ϵ)$ such that if a GSWF on three alternatives satisfies the IIA condition and its probability of...

On the uniform convergence of double sine series

Péter Kórus, Ferenc Móricz (2009)

Studia Mathematica

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Let a single sine series (*) $\sum_{k = 1}^{\infty} a_{k} s i n k x$ be given with nonnegative coefficients $a_{k}$ . If $a_{k}$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k a_{k} \to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) $\sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} c_{k l} s i n k x s i n l y$ , even with complex coefficients $c_{k l}$ . We also...

On sum-product representations in $ℤ_{q}$

Mei-Chu Chang (2006)

Journal of the European Mathematical Society

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The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$ , by performing sum and product set operations starting from a given subset $A$ of $ℤ_{q}$ . We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $| A | \sim log q$ ). Roughly speaking we show that all residue classes are obtained from a $k$ -fold sum of an $r$ -fold product set of $A$ , where $r ≪ log q$ and $log k ≪ log q$ , provided the...

Convolution operators with anisotropically homogeneous measures on $ℝ^{2 n}$ with n-dimensional support

E. Ferreyra, T. Godoy, M. Urciuolo (2002)

Colloquium Mathematicae

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Let $α_{i}, β_{i} > 0$ , 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t • x = (t^{α ₁} x ₁, . . ., t^{α ₙ} x ₙ)$ , $t \circ x = (t^{β ₁} x ₁, . . ., t^{β ₙ} x ₙ)$ and $| | x | | = \sum_{i = 1}^{n} {| x_{i} |}^{1 / α_{i}}$ . Let φ₁,...,φₙ be real functions in $C^{\infty} (ℝ ⁿ - 0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on $ℝ^{2 n}$ given by $μ (E) = \int_{ℝ ⁿ} χ_{E} {(x, φ (x)) | | x | |}^{γ - α} d x$ , where $α = \sum_{i = 1}^{n} α_{i}$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{μ} f = μ * f$ and let $| | T_{μ} {| |}_{p, q}$ be the operator norm of $T_{μ}$ from $L^{p} (ℝ^{2 n})$ into $L^{q} (ℝ^{2 n})$ , where the $L^{p}$ spaces are taken with respect to the Lebesgue measure. The type set $E_{μ}$ is defined by $E_{μ} = (1 / p, 1 / q) : | | T_{μ} {| |}_{p, q} < \infty, 1 \leq p, q \leq \infty$ . In the case $α_{i} \neq β_{k}$ for 1 ≤ i,k ≤ n we characterize the...