Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd Carl; David E. Edmunds

Displaying similar documents to “Gelfand numbers and metric entropy of convex hulls in Hilbert spaces”

Some logarithmic function spaces, entropy numbers, applications to spectral theory

Haroske Dorothee

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AbstractIn [18] and [19] we have studied compact embeddings of weighted function spaces on ℝⁿ, $i d : H_{q}^{s} (w (x), ℝ ⁿ) \to L ₚ (ℝ ⁿ)$ , s>0, 1 < q ≤ p< ∞, s-n/q+n/p > 0, with, for example, $w (x) = ⟨ x ⟩^{α}$ , α > 0, or $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0, and $⟨ x ⟩ = {(2 + | x | ²)}^{1 / 2}$ . We have determined the behaviour of their entropy numbers eₖ(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0. Our results in [18] imply that id cannot be compact for any β > 0, but after replacing the target space Lₚ(ℝⁿ) by a “slightly” larger one, $L ₚ {(l o g L)}_{- a} (ℝ ⁿ)$ , a...

ε-Entropy and moduli of smoothness in $L^{p}$ -spaces

A. Kamont (1992)

Studia Mathematica

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The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in $L^{p} (^{d})$ is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in $L^{p} (ℝ^{d})$ whose tail function decreases as $O (λ^{- γ})$ is obtained. In case p = 1 the relation between the ε-entropy of a given class of probability densities on $ℝ^{d}$ and the minimax risk for that class is discussed.

On the joint entropy of $d$ -wise-independent variables

Dmitry Gavinsky, Pavel Pudlák (2016)

Commentationes Mathematicae Universitatis Carolinae

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How low can the joint entropy of $n$ $d$ -wise independent (for $d \geq 2$ ) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$ , for $p < 1$ )? This question has been posed and partially answered in a recent work of Babai [Entropy versus pairwise independence (preliminary version), http://people.cs.uchicago.edu/ laci/papers/13augEntropy.pdf, 2013]. In this paper we improve some...

Jumps of entropy for $C^{r}$ interval maps

David Burguet (2015)

Fundamenta Mathematicae

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We study the jumps of topological entropy for $C^{r}$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^{r} ([0, 1])$ with $h_{t o p} (f) > (l o g ⁺ | | f^{'} {| |}_{\infty}) / r$ . To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^{r}$ interval maps.

Topological disjointness from entropy zero systems

Wen Huang, Kyewon Koh Park, Xiangdong Ye (2007)

Bulletin de la Société Mathématique de France

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The properties of topological dynamical systems $(X, T)$ which are disjoint from all minimal systems of zero entropy, $ℳ_{0}$ , are investigated. Unlike the measurable case, it is known that topological $K$ -systems make up a proper subset of the systems which are disjoint from $ℳ_{0}$ . We show that $(X, T)$ has an invariant measure with full support, and if in addition $(X, T)$ is transitive, then $(X, T)$ is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists...

On some nonlinear nonhomogeneous elliptic unilateral problems involving noncontrollable lower order terms with measure right hand side

C. Yazough, E. Azroul, H. Redwane (2013)

Applicationes Mathematicae

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We prove the existence of entropy solutions to unilateral problems associated to equations of the type $A u - d i v (ϕ (u)) = μ \in L ¹ (Ω) + W^{- 1, p^{'} (\cdot)} (Ω)$ , where A is a Leray-Lions operator acting from $W ₀^{1, p (\cdot)} (Ω)$ into its dual $W^{- 1, p (\cdot)} (Ω)$ and $ϕ \in C ⁰ (ℝ, ℝ^{N})$ .

On the directional entropy of ℤ²-actions generated by cellular automata

M. Courbage, B. Kamiński (2002)

Studia Mathematica

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We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule $F = F_{[l, r]}$ , l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy $h_{v ⃗} (Φ)$ , v⃗= (x,y) ∈ ℝ², is bounded above by $m a x (| z_{l} |, | z_{r} |) l o g A$ if $z_{l} z_{r} \geq 0$ and by $| z_{r} - z_{l} |$ in the opposite case, where $z_{l} = x + l y$ , $z_{r} = x + r y$ . We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.

Measures of maximal entropy for random $β$ -expansions

Karma Dajani, Martijn de Vries (2005)

Journal of the European Mathematical Society

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Let $β > 1$ be a non-integer. We consider $β$ -expansions of the form $\sum_{i = 1}^{\infty} d_{i} / β^{i}$ , where the digits ${(d_{i})}_{i \geq 1}$ are generated by means of a Borel map $K_{β}$ defined on ${0, 1}^{ℕ} \times [0, ⌊ β ⌋ / (β - 1)]$ . We show that $K_{β}$ has a unique mixing measure $ν_{β}$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure $ν_{β}$ the digits ${(d_{i})}_{i \geq 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness...

Hyperbolic measure of maximal entropy for generic rational maps of $ℙ^{k}$

Gabriel Vigny (2014)

Annales de l’institut Fourier

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Let $f$ be a dominant rational map of $ℙ^{k}$ such that there exists $s < k$ with $λ_{s} (f) > λ_{l} (f)$ for all $l$ . Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $Aut (ℙ^{k})$ , the map $f \circ A$ admits a hyperbolic measure of maximal entropy $log λ_{s} (f)$ with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of $f$ to $ℙ^{k + 1}$ . This provides many examples where non uniform hyperbolic dynamics is established. One of the key tools is to approximate...

Interaction between cellularity of Alexandroff spaces and entropy of generalized shift maps

Fatemah Ayatollah Zadeh Shirazi, Sahar Karimzadeh Dolatabad, Sara Shamloo (2016)

Commentationes Mathematicae Universitatis Carolinae

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In the following text for a discrete finite nonempty set $K$ and a self-map $ϕ : X \to X$ we investigate interaction between different entropies of generalized shift $σ_{ϕ} : K^{X} \to K^{X}$ , ${(x_{α})}_{α \in X} \mapsto {(x_{ϕ (α)})}_{α \in X}$ and cellularities of some Alexandroff topologies on $X$ .

$L^{2}$ -type contraction for systems of conservation laws

Denis Serre, Alexis F. Vasseur (2014)

Journal de l’École polytechnique — Mathématiques

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The semi-group associated with the Cauchy problem for a scalar conservation law is known to be a contraction in $L^{1}$ . However it is not a contraction in $L^{p}$ for any $p > 1$ . Leger showed in [] that for a convex flux, it is however a contraction in $L^{2}$ up to a suitable shift. We investigate in this paper whether such a contraction may happen for systems. The method is based on the relative entropy method. Our general analysis leads us to the new geometrical notion of systems. We treat in details...

Characteristic Exponents of Rational Functions

Anna Zdunik (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider two characteristic exponents of a rational function f:ℂ̂ → ℂ̂ of degree d ≥ 2. The exponent $χ_{a} (f)$ is the average of log∥f’∥ with respect to the measure of maximal entropy. The exponent $χ_{m} (f)$ can be defined as the maximal characteristic exponent over all periodic orbits of f. We prove that $χ_{a} (f) = χ_{m} (f)$ if and only if f(z) is conformally conjugate to $z \mapsto z^{\pm d}$ .

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

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For the real quadratic map $P_{a} (x) = x^{2} + a$ and a given $ϵ > 0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_{a}^{n} |_{J}$ univalent, with bounded distortion and $B (0, ϵ) \subseteq P_{a}^{n} (J)$ for some $n \in ℕ$ . The $ϵ$ -weakly expanding set is the set of points which do not have good expansion properties. Let $α$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[α, - α]$ . We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff...

Invariant densities for random $β$ -expansions

Karma Dajani, Martijn de Vries (2007)

Journal of the European Mathematical Society

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Let $β > 1$ be a non-integer. We consider expansions of the form $\sum_{i = 1}^{\infty} d_{i} / β^{i}$ , where the digits ${(d_{i})}_{i \geq 1}$ are generated by means of a Borel map $K_{β}$ defined on ${0, 1}^{ℕ} \times [0, ⌊ β ⌋ (β - 1)]$ . We show existence and uniqueness of a $K_{β}$ -invariant probability measure, absolutely continuous with respect to $m_{p} \otimes λ$ , where $m_{p}$ is the Bernoulli measure on ${0, 1}^{ℕ}$ with parameter $p$ ( $0 < p < 1$ ) and $λ$ is the normalized Lebesgue measure on $[0, ⌊ β ⌋ (β - 1)]$ . Furthermore, this measure is of the form $m_{p} \otimes μ_{β, p}$ , where $μ_{β, p}$ is equivalent to $λ$ . We prove that the measure of maximal entropy and $m_{p} \otimes λ$ are mutually...

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...

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Vitaly Moroz, Cyrill B. Muratov (2014)

Journal of the European Mathematical Society

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We study the leading order behaviour of positive solutions of the equation $- Δ u + ϵ u - {| u |}^{p - 2} u + {| u |}^{q - 2} u = 0, x \in ℝ^{N}$ , where $N \geq 3$ , $q > p > 2$ and when $ϵ > 0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$ , $q$ and $N$ . The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $2^{*} = \frac{2 N}{N - 2}$ . For $p < 2^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p > 2^{*}$ the solution asymptotically...

Equilibrium states for interval maps: the potential $- t log | D f |$

Henk Bruin, Mike Todd (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $f : I \to I$ be a $C^{2}$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential $φ_{t} : x \mapsto - t log | D f (x) |$ for $t$ close to $1$ , and also that the pressure function $t \mapsto P (φ_{t})$ is analytic on an appropriate interval near $t = 1$ .

Comparison of two methods for approximation of probability distributions with prescribed marginals

Albert Pérez, Milan Studený (2007)

Kybernetika

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Let $P$ be a discrete multidimensional probability distribution over a finite set of variables $N$ which is only partially specified by the requirement that it has prescribed given marginals ${P_{A}; A \in 𝒮}$ , where $𝒮$ is a class of subsets of $N$ with $⋃ 𝒮 = N$ . The paper deals with the problem of approximating $P$ on the basis of those given marginals. The divergence of an approximation $\hat{P}$ from $P$ is measured by the relative entropy $H (P | \hat{P})$ . Two methods for approximating $P$ are compared. One of them uses formerly introduced...

New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals

Yongsheng Han, Dachun Yang

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Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, ${(X, ϱ, μ)}_{d, θ}$ , which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces $B_{p q}^{s} (X)$ and Triebel-Lizorkin spaces $F_{p q}^{s} (X)$ have the lifting properties for |s| < θ. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional...

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...

Location of the critical points of certain polynomials

Somjate Chaiya, Aimo Hinkkanen (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $𝔻$ denote the unit disk ${z : | z | < 1}$ in the complex plane $ℂ$ . In this paper, we study a family of polynomials $P$ with only one zero lying outside $\bar{𝔻}$ . We establish criteria for $P$ to satisfy implying that each of $P$ and $P^{'}$ has exactly one critical point outside $\bar{𝔻}$ .