The max norm in -isometries and measure
Regina Cohen, James W. Fickett (1982)
Colloquium Mathematicae
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Regina Cohen, James W. Fickett (1982)
Colloquium Mathematicae
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Gogi Pantsulaia (2009)
Bulletin of the Polish Academy of Sciences. Mathematics
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New concepts of Lebesgue measure on 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 for α = (1,1,...).
Anders Johansson, Anders Öberg, Mark Pollicott (2012)
Journal of the European Mathematical Society
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We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique -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 -measure.
S. Lasher (1968)
Studia Mathematica
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Andrea R. Nahmod, Tadahiro Oh, Luc Rey-Bellet, Gigliola Staffilani (2012)
Journal of the European Mathematical Society
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We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier–Lebesgue space with and scaling like , for small . We also show the invariance of this measure.
Mrinal Kanti Roychowdhury (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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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 of ν and bounded above by a unique number , related to the temperature function of the thermodynamic formalism that arises in the multifractal analysis of μ.
Eric Amar (2008)
Annales Polonici Mathematici
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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 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 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...
Uffe Haagerup, Hanne Schultz (2009)
Publications Mathématiques de l'IHÉS
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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 affiliated with ℳ, such that the Brown measure of is concentrated...
Valentino Magnani (2006)
Journal of the European Mathematical Society
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We establish an explicit connection between the perimeter measure of an open set with boundary and the spherical Hausdorff measure restricted to , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of is less than or equal to up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...
Stanisaw Szufla (1998)
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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We present a new theorem on the differential inequality . Next, we apply this result to obtain existence theorems for the equation .
G. Pantsulaia (2004)
Bulletin of the Polish Academy of Sciences. Mathematics
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An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift of μ by a vector are neither equivalent nor orthogonal. This extends a result established in [7].
(2014)
Acta Arithmetica
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We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure of a polynomial where is the integral of over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding , in particular as k → ∞.
Márton Elekes, Juris Steprāns (2004)
Fundamenta Mathematicae
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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 ) that less than many translates of a compact set of measure zero can cover ℝ.
Julien Melleray (2014)
Annales de l’institut Fourier
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We show that, whenever is a countable abelian group and is a finitely-generated subgroup of , a generic measure-preserving action of on a standard atomless probability space extends to a free measure-preserving action of on . This extends a result of Ageev, corresponding to the case when is infinite cyclic.
Luis Bernal-González (2010)
Studia Mathematica
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We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological...
G. Pantsulaia (2004)
Bulletin of the Polish Academy of Sciences. Mathematics
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An example of a non-zero non-atomic translation-invariant Borel measure on the Banach space is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition "-almost every element of has a property P" implies that “almost every” element of (in the sense of [4]) has the property P. It is also shown that the converse is not valid.
John Garza (2014)
Acta Arithmetica
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For an algebraic number field and a subset , we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in vanishing at the point .
Jozef Myjak, Tomasz Szarek (2002)
Annales Polonici Mathematici
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We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems where are bi-lipschitzean transformations.
Mrinal Kanti Roychowdhury, Daniel J. Rudolph (2008)
Fundamenta Mathematicae
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Two invertible dynamical systems (X,,μ,T) and (Y,,ν,S), where X and Y are Polish spaces and Borel probability spaces and T, S are measure preserving homeomorphisms of X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X₀ of X of measure one onto a subset Y₀ of Y of full measure such that (1) is continuous in the relative topology on X₀ and is continuous in the relative topology on Y₀, (2) for μ-a.e. x ∈ X. (X,,μ,T)...
Przemysław Liszka (2013)
Annales Polonici Mathematici
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Let for i = 1,..., N be contracting similarities, let be a probability vector and let ν be a probability measure on with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on such that . We give satisfactory estimates for the lower and upper bounds of the spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...
Tom Kempton (2016)
Journal of the European Mathematical Society
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We study natural measures on sets of -expansions and on slices through self similar sets. In the setting of -expansions, these allow us to better understand the measure of maximal entropy for the random -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,...
Fan Lü, Bo Tan, Jun Wu (2014)
Fundamenta Mathematicae
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For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with . We prove that for any x ∈ (0,1), (x) contains a sequence increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure are nowhere dense.
Nijjwal Karak (2017)
Czechoslovak Mathematical Journal
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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point in a metric measure space is called a generalized Lebesgue point of a measurable function if the medians of over the balls converge to when converges to . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....
Geraldo Botelho, Daniel Cariello, Vinícius V. Fávaro, Daniel Pellegrino, Juan B. Seoane-Sepúlveda (2013)
Studia Mathematica
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Tatsuhiko Yagasaki (2007)
Fundamenta Mathematicae
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Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism , which measures for each the mass flow toward ends induced by h. We show that the...
Karma Dajani, Martijn de Vries (2007)
Journal of the European Mathematical Society
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Let be a non-integer. We consider expansions of the form , where the digits are generated by means of a Borel map defined on . We show existence and uniqueness of a -invariant probability measure, absolutely continuous with respect to , where is the Bernoulli measure on with parameter () and is the normalized Lebesgue measure on . Furthermore, this measure is of the form , where is equivalent to . We prove that the measure of maximal entropy and are mutually...
S. Okada, W. J. Ricker, E. A. Sánchez Pérez
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The spaces L¹(m) of all m-integrable (resp. of all scalarly m-integrable) functions for a vector measure m, taking values in a complex locally convex Hausdorff space X (briefly, lcHs), are themselves lcHs for the mean convergence topology. Additionally, is always a complex vector lattice; this is not necessarily so for L¹(m). To identify precisely when L¹(m) is also a complex vector lattice is one of our central aims. Whenever X is sequentially complete, then this is the case. If,...
Andrea Carbonaro, Giancarlo Mauceri, Stefano Meda (2010)
Colloquium Mathematicae
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In a previous paper the authors developed an H¹-BMO theory for unbounded metric measure spaces (M,ρ,μ) of infinite measure that are locally doubling and satisfy two geometric properties, called “approximate midpoint” property and “isoperimetric” property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies...
Abdessatar Barhoumi, Habib Ouerdiane, Anis Riahi (2007)
Banach Center Publications
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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 , 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 by using the so-called β-type Wick product.
Ryotaro Sato (2004)
Colloquium Mathematicae
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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 in measure for some f ∈ M(μ;X), then also 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)...
Roland Zweimüller (2002)
Colloquium Mathematicae
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We construct 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 of arithmetical averages of image measures does not converge weakly.