On Peano derivatives in
S. Lasher (1968)
Studia Mathematica
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S. Lasher (1968)
Studia Mathematica
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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.
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...
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,...
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.
(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 → ∞.
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...
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.
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.
Yu Deng (2015)
Journal of the European Mathematical Society
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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 , extending the well-posedness result of Molinet [20].
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].
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 .
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...
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....
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.
Geraldo Botelho, Daniel Cariello, Vinícius V. Fávaro, Daniel Pellegrino, Juan B. Seoane-Sepúlveda (2013)
Studia Mathematica
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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 .
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,...
Alessio Figalli, David Jerison (2015)
Journal of the European Mathematical Society
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Given a measurable set of positive measure, it is not difficult to show that if and only if is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If is small, is close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between and its convex hull in terms of .
Teresa Bermúdez, Carlos Díaz Mendoza, Antonio Martinón (2012)
Studia Mathematica
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A bounded linear operator T on a Banach space X is called an (m,p)-isometry for a positive integer m and a real number p ≥ 1 if, for any vector x ∈ X, . We prove that any power of an (m,p)-isometry is also an (m,p)-isometry. In general the converse is not true. However, we prove that if and are (m,p)-isometries for a positive integer r, then T is an (m,p)-isometry. More precisely, if is an (m,p)-isometry and is an (l,p)-isometry, then is an (h,p)-isometry, where t = gcd(r,s)...
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)...
Zbigniew Lipecki (2015)
Commentationes Mathematicae Universitatis Carolinae
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Let and be algebras of subsets of a set with , and denote by the set of all quasi-measure extensions of a given quasi-measure on to . We give some criteria for order boundedness of in , in the general case as well as for atomic . Order boundedness implies weak compactness of . We show that the converse implication holds under some assumptions on , and or alone, but not in general.
Simon Baker (2015)
Acta Arithmetica
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Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence a β-expansion for x if . We call a finite sequence an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given , we introduce the following subset of ℝ: In other words, is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities . When , the Borel-Cantelli lemma tells us that the Lebesgue measure...