On translations of subsets of the real line

In this paper we discuss various questions connected with translations of subsets of the real line. Most of these questions originate from W. Sierpinski. We discuss the number of translations a single subset of the reals may have. Later we discuss almost invariant subsets of Abelian groups.

     

Remarks on nonmeasurable unions of big point families

We show that under some conditions on a family A ? I there exists a subfamily A₀ ? A such that ∪ A₀ is nonmeasurable with respect to a fixed ideal I with Borel base of a fixed uncountable Polish space. Our result applies to the classical ideal of null subsets of the real line and to the ideal of first category subsets of the real line (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuous transformation of Baire measures into Lebesgue measure

A recent result by Wulbert on the existence of continuous functions with measure zero level sets is slightly extended and its proof is considerably simplified. As a by-product, a criterion is established for a Baire measure to allow a continuous transformation into Lebesgue measure.

     

On the measure theoretic structure of compact groups

If is a compact group with , we show the following results:

(i)

There exist direct products of compact metric groups and continuous open surjections with respect to Haar measure; and

(ii)

the Haar measure on is Baire and at the same time Jordan isomorphic to the Haar measure on a direct product of compact Lie groups.

Applications of the above results in measure theory are given.

     

Existence of EF‐equivalent non‐isomorphic models

We prove the existence of pairs of models of the same cardinality λ which are very equivalent according to EF games, but not isomorphic. We continue the paper [4], but we do not rely on it. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Carleson measures and conformal self‐mappings in the real unit ball

Bounded and compact Carleson measures in the unit ball B of R ⁿ , n ≥ 2, are characterized by means of global Dirichlet integrals of the conformal self‐map T_a taking a ∈ B to the origin. The same proof applies in the unit ball of C ⁿ . It is also proved that the powers of the Jacobian of T_a satisfy the weak Harnack inequality and even Harnack's inequality with a constant independent of a. As an application of these results it is shown that the two different definitions for Carleson measures in the existing literature are equivalent for a certain range of parameter values. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Uncountable Unions and Intersections of Measurable Sets

We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a good additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are considered in detail, and it is shown that the Lebesgue measure and the Baire property differ essentially in this aspect. Another difference between measure and category is illustrated in the case of some uncountable intersections of sets of full measure (comeager sets, respectively). We also discuss a topological form of the Vitali covering theorem, in connection with the Baire property of uncountable unions of certain sets.     

On the predictability of discrete dynamical systems

Let be a metric space. A function is said to be non-sensitive at a point if for every 0$"> there is a 0$"> such that for any choice of points , , , we have that for every . Let be the set of all homeomorphisms from onto endowed with the topology of uniform convergence. The main goal of the present paper is to prove that for certain spaces , ``most' functions in are non-sensitive at ``most' points of .

     

On Lebesgue Functions of Uniformly Bounded Orthonormal Systems

A complement to A. M. Olevskii's fundamental inequality on the logarithmic growth of Lebesgue functions of an arbitrary uniformly bounded orthonormal system on a set of positive measure is made. Namely, the index where the Lebesgue functions have growth slightly weaker than logarithmic can be chosen independently of the variable. The theorem proved in this paper improves a result established earlier by the author.     

The size of the Dini subdifferential

Given a lower semicontinuous function , we prove that the points of , where the lower Dini subdifferential contains more than one element, lie in a countable union of sets which are isomorphic to graphs of some Lipschitzian functions defined on . Consequently, the set of all these points has a null Lebesgue measure.

     

Classification of non‐well‐founded sets and an application

A complete list of Finsler, Scott and Boffa sets whose transitive closures contain 1, 2 and 3 elements is given. An algorithm for deciding the identity of hereditarily finite Scott sets is presented. Anti‐well‐founded (awf) sets, i. e., non‐well‐founded sets whose all maximal ∈‐paths are circular, are studied. For example they form transitive inner models of ZFC minus foundation and empty set, and they include uncountably many hereditarily finite awf sets. A complete list of Finsler and Boffa awf sets with 2 and 3 elements in their transitive closure is given. Next the existence of infinite descending ∈‐sequences in Aczel universes is shown. Finally a theorem of Ballard and Hrbá?ek concerning nonstandard Boffa universes of sets is considerably extended.     

函数π(x;α,β)的单调性

在文章中，我们对Gerber(1992)所提出的与函数π（x;α,β)相关的单调性问题给出了一个分析的证明。     

Quadratic functional equations in a set of Lebesgue measure zero

Let R$\mathbb{R}$ be the set of real numbers, Y   a Banach space and f:R→Y$f:\mathbb{R}\to Y$. We prove the Hyers–Ulam stability theorem for the quadratic functional inequality

‖f(x+y)+f(x−y)−2f(x)−2f(y)‖≤?$\Vert f(x+y)+f(x-y)-2f(x)-2f(y)\Vert \le ?$

for all (x,y)∈Ω$(x,y)\in \Omega$, where Ω⊂R²$\Omega \subset {\mathbb{R}}^2$ is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0.     

Some results on projection of planar sets

In this paper we define certain types of projections of planar sets and study some properties of such projections.     

Choice principles for special subsets of the real line

We study the role the axiom of choice plays in the existence of some special subsets of ? and its power set ?(?).     

Self‐organized criticality of cellular automata model; absorbtion in finite‐time of supercritical region into the critical one

In this work, it is studied the evolution and time behavior of solutions to nonlinear diffusion equation in where , d ≥ 1, and H is the Heaviside function. For d = 1,2,3, this equation describes the dynamics of self‐organizing sandpile process with critical state ρ_c. The main conclusion is that the supercritical region is absorbed in a finite‐time in the critical region . Copyright © 2013 John Wiley & Sons, Ltd.     

Non‐standard finite difference schemes for multi‐dimensional second‐order systems in non‐smooth mechanics

This work is an extension of the paper (Proc. R. Soc. London 2005; 461A :1927–1950) to impact oscillators with more than one degree of freedom. Given the complex and even chaotic behaviour of these non‐smooth mechanical systems, it is essential to incorporate their qualitative physical properties, such as the impact law and the frequencies of the systems, into the envisaged numerical methods if the latter is to be reliable. Based on this strategy, we design several non‐standard finite difference schemes. Apart from their excellent error bounds and unconditional stability, the schemes are analysed for their efficiency to preserve some important physical properties of the systems including, among others, the conservation of energy between consecutive impact times, the periodicity of the motion and the boundedness of the solutions. Numerical simulations that support the theory are provided. Copyright © 2006 John Wiley & Sons, Ltd.