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Entropy on effect algebras with the Riesz decomposition property I: Basic properties

Antonio Di Nola, Anatolij Dvurečenskij, Marek Hyčko, Corrado Manara (2005)

Kybernetika

We define the entropy, lower and upper entropy, and the conditional entropy of a dynamical system consisting of an effect algebra with the Riesz decomposition property, a state, and a transformation. Such effect algebras allow many refinements of two partitions. We present the basic properties of these entropies and these notions are illustrated by many examples. Entropy on MV-algebras is postponed to Part II.

Equations containing locally Henstock-Kurzweil integrable functions

Seppo Heikkilä, Guoju Ye (2012)

Applications of Mathematics

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.

Equidecomposability of Jordan domains under groups of isometries

M. Laczkovich (2003)

Fundamenta Mathematicae

Let G d denote the isometry group of d . We prove that if G is a paradoxical subgroup of G d then there exist G-equidecomposable Jordan domains with piecewise smooth boundaries and having different volumes. On the other hand, we construct a system d of Jordan domains with differentiable boundaries and of the same volume such that d has the cardinality of the continuum, and for every amenable subgroup G of G d , the elements of d are not G-equidecomposable; moreover, their interiors are not G-equidecomposable...

Equilibrium measures for holomorphic endomorphisms of complex projective spaces

Mariusz Urbański, Anna Zdunik (2013)

Fundamenta Mathematicae

Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space k , k ≥ 1, and let J be the Julia set of f (the topological support of the unique maximal entropy measure). Then there exists a positive number κ f > 0 such that if ϕ: J → ℝ is a Hölder continuous function with s u p ( ϕ ) - i n f ( ϕ ) < κ f , then ϕ admits a unique equilibrium state μ ϕ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system ( f , μ ϕ ) is K-mixing, whence ergodic. Proving...

Equivalent or absolutely continuous probability measures with given marginals

Patrizia Berti, Luca Pratelli, Pietro Rigo, Fabio Spizzichino (2015)

Dependence Modeling

Let (X,A) and (Y,B) be measurable spaces. Supposewe are given a probability α on A, a probability β on B and a probability μ on the product σ-field A ⊗ B. Is there a probability ν on A⊗B, with marginals α and β, such that ν ≪ μ or ν ~ μ ? Such a ν, provided it exists, may be useful with regard to equivalent martingale measures and mass transportation. Various conditions for the existence of ν are provided, distinguishing ν ≪ μ from ν ~ μ.

Equivanishing sequences of mappings

Piotr Antosik (2000)

Banach Center Publications

Utilizing elementary properties of convergence of numerical sequences we prove Nikodym, Banach, Orlicz-Pettis type theorems

Equivariant measurable liftings

Nicolas Monod (2015)

Fundamenta Mathematicae

We discuss equivariance for linear liftings of measurable functions. Existence is established when a transformation group acts amenably, as e.g. the Möbius group of the projective line. Since the general proof is very simple but not explicit, we also provide a much more explicit lifting for semisimple Lie groups acting on their Furstenberg boundary, using unrestricted Fatou convergence. This setting is relevant to L -cocycles for characteristic classes.

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