On some results about convex functions of order
M. Obradović, S. Owa (1986)
Matematički Vesnik
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M. Obradović, S. Owa (1986)
Matematički Vesnik
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Wojciech Kosek (2011)
Colloquium Mathematicae
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The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with . In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that In this note we construct an example of a nonpositive L¹ operator with the...
Philippe Laurençot (2002)
Colloquium Mathematicae
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If φ: [0,∞) → [0,∞) is a convex function with φ(0) = 0 and conjugate function φ*, the inequality is shown to hold true for every ε ∈ (0,∞) if and only if φ* satisfies the Δ₂-condition.
Leonid Polterovich (1999)
Journal of the European Mathematical Society
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Let be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold . A loop is called strictly ergodic if for some irrational number the associated skew product map defined by is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply...
Alberto Seeger (1997)
Annales Polonici Mathematici
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Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family involves the concept of cumulant transformation and a standard homogenization procedure.
Stefan Müller, Vladimír Šverák (1999)
Journal of the European Mathematical Society
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We study solutions of first order partial differential relations , where is a Lipschitz map and is a bounded set in matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of and second we replace Gromov’s −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our...
Prakash G. Umarani (1983)
Annales Polonici Mathematici
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Isaac Kornfeld, Wojciech Kosek (2003)
Colloquium Mathematicae
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E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, ). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0A class of numerical sequences αₙ, intimately...
Jean-Pierre Conze, Albert Raugi (2009)
Colloquium Mathematicae
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Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure . We consider the map defined on X × G by and the cocycle generated by φ. Using a characterization of the ergodic invariant measures for , we give the form of the ergodic decomposition of or more generally of the -invariant measures , where is χ∘φ-conformal for an exponential χ on G.
Christophe Cuny (2010)
Studia Mathematica
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Let X be a closed subspace of , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like , 0 ≤ α < 1, in terms of , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...
Laura Burlando (2004)
Studia Mathematica
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An improvement of the generalization-obtained in a previous article [Bu1] by the author-of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Namely, two bounded linear operators T and A are constructed such that converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and converges uniformly, the sum of the range of 1-A and the kernel of (1-A)²...
David Färm, Tomas Persson (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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We study sets of non-typical points under the map mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β > 1.541 found in the above paper can be removed.
Bo’az Klartag (2013)
Annales de la faculté des sciences de Toulouse Mathématiques
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We discuss a method for obtaining Poincaré-type inequalities on arbitrary convex bodies in . Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also applies to some non-convex sets. In particular, we generalize the central limit theorem for convex bodies to a class of non-convex domains, including the unit balls of -spaces in for .
Patrick LaVictoire (2011)
Colloquium Mathematicae
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We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence such that the weighted ergodic averages corresponding to satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions,...
Françoise Lust-Piquard (1989)
Colloquium Mathematicae
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Grzegorz Lewicki, Michael Prophet (2007)
Studia Mathematica
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We say that a function from is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general “shape”...
Vladimir Fonf, Menachem Kojman (2001)
Fundamenta Mathematicae
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We investigate countably convex subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets...
Fethi Kadhi (2002)
RAIRO - Operations Research - Recherche Opérationnelle
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We investigate the minima of functionals of the form where is strictly convex. The admissible functions are not necessarily convex and satisfy on , , , is a fixed function on . We show that the minimum is attained by , the convex envelope of .
Rui Kuang, Xiangdong Ye (2008)
Colloquium Mathematicae
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In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any...
B. Mirković (1970)
Matematički Vesnik
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Joanna Cyman, Magdalena Lemańska, Joanna Raczek (2006)
Discussiones Mathematicae Graph Theory
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For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number of a...
Elmouloudi Ed-dari (2004)
Studia Mathematica
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For a given linear operator T in a complex Banach space X and α ∈ ℂ with ℜ (α) > 0, we define the nth Cesàro mean of order α of the powers of T by . For α = 1, we find , the usual Cesàro mean. We give necessary and sufficient conditions for a (C,α) bounded operator to be (C,α) strongly (weakly) ergodic.
A.P. Santhakumaran, S.V. Ullas Chandran (2012)
Discussiones Mathematicae Graph Theory
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For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), . A set S of vertices is a detour convex set if . The detour convex hull is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of...
Guy Cohen, Christophe Cuny, Michael Lin (2010)
Studia Mathematica
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Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform . We prove that weak and strong convergence are equivalent, and in a reflexive space also is equivalent to the convergence. We also show that (which converges on (I-T)X) is precisely the infinitesimal generator of the semigroup .
Katsuro Sakai, Zhongqiang Yang (2007)
Bulletin of the Polish Academy of Sciences. Mathematics
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Let be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that for every n > 1 whereas .
S. Owa, C. Y. Shen (1988)
Matematički Vesnik
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