On Measure Concentration of Vector-Valued Maps

Michel Ledoux; Krzysztof Oleszkiewicz

The search session has expired. Please query the service again.

Displaying similar documents to “On Measure Concentration of Vector-Valued Maps”

$α$ -continuous multivalued maps

J. Eweret (1989)

Matematički Vesnik

Similarity:

Exact $^{\infty}$ covering maps of the circle without (weak) limit measure

Roland Zweimüller (2002)

Colloquium Mathematicae

Similarity:

We construct $^{\infty}$ maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ${(n^{- 1} \sum_{k = 0}^{n - 1} ν \circ T^{- k})}_{n \geq 1}$ of arithmetical averages of image measures does not converge weakly.

Unique Bernoulli $g$ -measures

Anders Johansson, Anders Öberg, Mark Pollicott (2012)

Journal of the European Mathematical Society

Similarity:

We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a $g$ -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique $g$ -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the $g$ -measure.

The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

Similarity:

Invariance of the Gibbs measure for the Benjamin–Ono equation

Yu Deng (2015)

Journal of the European Mathematical Society

Similarity:

In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than $L^{2}$ , extending the $L^{2}$ well-posedness result of Molinet [20].

On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

New concepts of Lebesgue measure on $ℝ^{\infty}$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on $ℝ^{\infty}$ for α = (1,1,...).

Infinite dimensional Gegenbauer functionals

Abdessatar Barhoumi, Habib Ouerdiane, Anis Riahi (2007)

Banach Center Publications

Similarity:

he paper is devoted to investigation of Gegenbauer white noise functionals. A particular attention is paid to the construction of the infinite dimensional Gegenbauer white noise measure $_{β}$ , via the Bochner-Minlos theorem, on a suitable nuclear triple. Then we give the chaos decomposition of the L²-space with respect to the measure $_{β}$ by using the so-called β-type Wick product.

Quantization Dimension Estimate of Inhomogeneous Self-Similar Measures

Mrinal Kanti Roychowdhury (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We consider an inhomogeneous measure μ with the inhomogeneous part a self-similar measure ν, and show that for a given r ∈ (0,∞) the lower and the upper quantization dimensions of order r of μ are bounded below by the quantization dimension $D_{r} (ν)$ of ν and bounded above by a unique number $κ_{r} \in (0, \infty)$ , related to the temperature function of the thermodynamic formalism that arises in the multifractal analysis of μ.

Solvability of the functional equation f = (T-I)h for vector-valued functions

Ryotaro Sato (2004)

Colloquium Mathematicae

Similarity:

Let X be a reflexive Banach space and (Ω,,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and $l i m_{n \to \infty} f ₙ = f$ in measure for some f ∈ M(μ;X), then also $l i m_{n \to \infty} T f ₙ = T f$ in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X)...

Some remarks providing discontinuous maps on some $C_{p} (X)$ spaces

S. Moll (2008)

Banach Center Publications

Similarity:

Let X be a completely regular Hausdorff topological space and $C_{p} (X)$ the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space $C_{p} (X)$ , if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of $C_{p} (X)$ .

Asymptotic behaviour of averages of k-dimensional marginals of measures on ℝⁿ

Jesús Bastero, Julio Bernués (2009)

Studia Mathematica

Similarity:

We study the asymptotic behaviour, as n → ∞, of the Lebesgue measure of the set $x \in K : | P_{E} (x) | \leq t$ for a random k-dimensional subspace E ⊂ ℝⁿ and an isotropic convex body K ⊂ ℝⁿ. For k growing slowly to infinity, we prove it to be close to the suitably normalised Gaussian measure in $ℝ^{k}$ of a t-dilate of the Euclidean unit ball. Some of the results hold for a wider class of probabilities on ℝⁿ.

On Peano derivatives in $L^{p} (E_{n})$

S. Lasher (1968)

Studia Mathematica

Similarity:

On an Invariant Borel Measure in Hilbert Space

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold $_{μ}$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift $μ^{(a)}$ of μ by a vector $a \in ℓ ₂ ∖_{μ}$ are neither equivalent nor orthogonal. This extends a result established in [7].

Lipschitz extensions of convex-valued maps

Alberto Bressan, Agostino Cortesi (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz $M$ , definita su un sottoinsieme di uno spazio di Hilbert $H$ a valori compatti e convessi in $\mathbb{R}^{n}$ , può essere estesa su tutto $H$ ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz $M$ .

Relaxation theorem for set-valued functions with decomposable values

Andrzej Kisielewicz (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

Let (T,F,μ) be a separable probability measure space with a nonatomic measure μ. A subset K ⊂ L(T,Rⁿ) is said to be decomposable if for every A ∈ F and f ∈ K, g ∈ K one has $f χ_{A} + g χ_{T} \in K$ . Using the property of decomposability as a substitute for convexity a relaxation theorem for fixed point sets of set-valued function is given.

Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

Piotr Niemiec (2009)

Studia Mathematica

Similarity:

It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $C F L (_{r})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space $_{r}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $L I P (_{r})$ of equivalence classes of all real-valued Lipschitz maps on $_{r}$ . The space of all signed (real-valued) Borel measures on $_{r}$ is isometrically embedded in the dual space of $C F L (_{r})$ and it is shown that the image...

Interpolating sequences, Carleson measures and Wirtinger inequality

Eric Amar (2008)

Annales Polonici Mathematici

Similarity:

Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure $μ_{S} : = \sum_{a \in S} (1 - | a | ²) ⁿ δ_{a}$ is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure $μ_{S}$ bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual...

Interpolation and extension of Lipschitz-Hölder maps on $C_{p}$ spaces

Charles E. C Cleaver (1973)

Colloquium Mathematicae

Similarity:

Sets of $β$ -expansions and the Hausdorff measure of slices through fractals

Tom Kempton (2016)

Journal of the European Mathematical Society

Similarity:

We study natural measures on sets of $β$ -expansions and on slices through self similar sets. In the setting of $β$ -expansions, these allow us to better understand the measure of maximal entropy for the random $β$ -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing,...

Less than $2^{ω}$ many translates of a compact nullset may cover the real line

Márton Elekes, Juris Steprāns (2004)

Fundamenta Mathematicae

Similarity:

We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from $c o f () < 2^{ω}$ ) that less than $2^{ω}$ many translates of a compact set of measure zero can cover ℝ.

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila, Mikhail Lyubich, Weixiao Shen (2011)

Journal of the European Mathematical Society

Similarity:

We study the parameter space of unicritical polynomials $f_{c} : z \mapsto z^{d} + c$ . For complex parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

Similarity:

Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦 = 𝒦_{T} (B)$ affiliated with ℳ, such that the Brown measure of ${T |}_{𝒦}$ is concentrated...

Note on the variance of the sum of Gaussian functionals

Marek Beśka (2010)

Applicationes Mathematicae

Similarity:

Let $(X_{i}, i = 1, 2, . . .)$ be a Gaussian sequence with $X_{i} \in N (0, 1)$ for each i and suppose its correlation matrix $R = {(ρ_{i j})}_{i, j \geq 1}$ is the matrix of some linear operator R:l₂→ l₂. Then for $f_{i} \in L ² (μ)$ , i=1,2,..., where μ is the standard normal distribution, we estimate the variation of the sum of the Gaussian functionals $f_{i} (X_{i})$ , i=1,2,... .

Distinguished subspaces of $L_{p}$ of maximal dimension

Geraldo Botelho, Daniel Cariello, Vinícius V. Fávaro, Daniel Pellegrino, Juan B. Seoane-Sepúlveda (2013)

Studia Mathematica

Similarity:

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

We present a new theorem on the differential inequality $u^{(m)} \leq w (u)$ . Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f (t, x)$ .

Local analysis of a cubically convergent method for variational inclusions

Steeve Burnet, Alain Pietrus (2011)

Applicationes Mathematicae

Similarity:

This paper deals with variational inclusions of the form 0 ∈ φ(x) + F(x) where φ is a single-valued function admitting a second order Fréchet derivative and F is a set-valued map from $ℝ^{q}$ to the closed subsets of $ℝ^{q}$ . When a solution z̅ of the previous inclusion satisfies some semistability properties, we obtain local superquadratic or cubic convergent sequences.

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

Similarity:

We establish an explicit connection between the perimeter measure of an open set $E$ with $C^{1}$ boundary and the spherical Hausdorff measure $S^{Q - 1}$ restricted to $\partial E$ , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and $Q$ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of $E$ is less than or equal to $S^{Q - 1} (\partial E)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

The Bohr-Pál theorem and the Sobolev space $W ₂^{1 / 2}$

Vladimir Lebedev (2015)

Studia Mathematica

Similarity:

The well-known Bohr-Pál theorem asserts that for every continuous real-valued function f on the circle there exists a change of variable, i.e., a homeomorphism h of onto itself, such that the Fourier series of the superposition f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space $W ₂^{1 / 2} ()$ . This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if...

Lipschitz extensions of convex-valued maps

Alberto Bressan, Agostino Cortesi (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity: