A theorem on ROD‐hypersmooth equivalence relations in the Solovay model

It is known that every Borel hypersmooth but non‐smooth equivalence relation is Borel bi‐reducible to E₁. We prove a ROD version of this result in the Solovay model.     

The r-Rank of the groups of exceptional Lie type

In this note, we prove the following result, settling a question raised at the end of [Borel & Serre 1953], cf. [Borel 1983 pp. 228 and 708]. A related result for Lie groups of type E₈ was recently proved by J.F. Adams.     

"Whitney引理的推广及应用"一文中的错误

本文用一个十分简单的例子说明[1]对整体的Borel定理的证明是错误的.为此, 还须介绍函数芽和函数芽序列一致收敛的概念,并给出一个判定引理.     

Effective Borel degrees of some topological functions

The focus of this paper is the incomputability of some topological functions (with respect to certain representations) using the tools of Borel computability theory, as introduced by V. Brattka in [3] and [4]. First, we analyze some basic topological functions on closed subsets of ?ⁿ , like closure, border, intersection, and derivative, and we prove for such functions results of Σ⁰₂‐completeness and Σ⁰₃‐completeness in the effective Borel hierarchy. Then, following [13], we re‐consider two well‐known topological results: the lemmas of Urysohn and Urysohn‐Tietze for generic metric spaces (for the latter we refer to the proof given by Dieudonné). Both lemmas define Σ⁰₂‐computable functions which in some cases are even Σ⁰₂‐complete. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

First Borel class sets in Banach spaces and the asymptotic-norming property

The Radon-Nikodym property in a separable Banach spaceX is related to the representation ofX as a weak* first Borel class subset of some dual Banach space (its bidualX**, for instance) by well known results due to Edgar and Wheeler [8], and Ghoussoub and Maurey [9, 10, 11]. The generalizations of those results depend on a new notion of Borel set of the first class “generated by convex sets” which is more suitable to deal with non-separable Banach spaces. The asymptotic-norming property, introduced by James and Ho [13], and the approximation by differences of convex continuous functions are also studied in this context. Research partially supported by the grant DGES PB 98-0381.     

Beyond Borel-amenability: Scales and superamenable reducibilities

We analyze the degree-structure induced by large reducibilities under the Axiom of Determinacy. This generalizes the analysis of Borel reducibilities given in Alessandro Andretta and Donald A. Martin (2003) [1], Luca Motto Ros (2009) [6] and Luca Motto Ros. (in press) [5] e.g. to the projective levels.     

On the Borel property for solutions to systems of complex vector fields

In this work we study the Borel property for smooth solutions to systems of complex vector fields associated to locally integrable structures. Inspired by the recent article [6], in which the Borel property was studied for generic submanifolds of the complex space, we prove similar results in this more general set up. In particular we obtain, for the case of corank one structures, a necessary and sufficient condition for the validity of the Borel property.     

No Borel Connections for the Unsplitting Relations

We prove that there is no Borel connection for non‐trivial pairs of unsplitting relations. This was conjectured in [3].     

On the equality of two-variable general functional means

Aequationes mathematicae - Given two functions $$f,g:I\rightarrow \mathbb {R}$$ and a probability measure $$\mu $$ on the Borel subsets of [0, 1], the two-variable mean $$M_{f,g;\mu...     

Some applications of the Adams-Kechris technique

We analyze the technique used by Adams and Kechris (2000) to obtain their results about Borel reducibility of countable Borel equivalence relations. Using this technique, we show that every equivalence relation is Borel reducible to the Borel bi-reducibility of countable Borel equivalence relations. We also apply the technique to two other classes of essentially uncountable Borel equivalence relations and derive analogous results for the classification problem of Borel automorphisms.

     

Abelian pro‐countable groups and non‐Borel orbit equivalence relations

We study topological groups that can be defined as Polish, pro‐countable abelian groups, as non‐archimedean abelian groups or as quasi‐countable abelian groups, i.e., Polish subdirect products of countable, discrete groups, endowed with the product topology. We characterize tame groups in this class, i.e., groups all of whose continuous actions on a Polish space induce a Borel orbit equivalence relation, and relatively tame groups, i.e., groups all of whose diagonal actions induce a Borel orbit equivalence relation, provided that are continuous actions inducing Borel orbit equivalence relations.     

ON THE DISTRIBUTION OF VALUES OF RANDOM DIRICHLET SERIES(II)

For certain Dirichlet series almost surely(a.s.)of order(R) ρ∈(0,∞)in the right-halfplane,a.s.every point of the imaginary axis is a Borel point of order ρ+1 and with no finiteexoeptional value.     

Borel sets and functions in topological spaces

We present a construction of the Borel hierarchy in general topological spaces and its relation to Baire hierarchy. We define mappings of Borel class α, prove the validity of the Lebesgue-Hausdor-Banach characterization for them and show their relation to Baire classes of mappings on compact spaces. The obtained results are used for studying Baire and Borel order of compact spaces, answering thus one part of a question raised by R. D. Mauldin. We present several examples showing some natural limits of our results in non-compact spaces.     

GENERATING SEQUENCES OF FUNCTIONS

We consider the problem of obtaining an arbitrary countablecollection of functions with specific properties as a compositionof finitely many functions with the same property. The functionsinvestigated are continuous, Baire-n, Lebesgue or Borel measurable,increasing, and differentiable functions on [0, 1], and increasingfunctions on .     

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF -algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.

     

Isomorphism of subshifts is a universal countable borel equivalence relation

We use the theory of Borel equivalence relations to analyze the equivalence relation of isomorphism among one-dimensional subshifts. We show that this equivalence relation is a universal countable Borel equivalence relation, so that it admits no definable complete invariants fundamentally simpler than the equivalence classes. We also see that the classification of higher dimensional subshifts up to isomorphism has the same complexity as for the one-dimensional case.     

Effective Borel measurability and reducibility of functions

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single‐valued as well as for multi‐valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya‐Srivastava in order to prove a generalization of the Representation Theorem of Kreitz‐Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach‐Hausdorff‐Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Borel on the Questions Versus Borel on the Answers

We consider morphisms (also called Galois-Tukey connections) between binary relations that are used in the theory of cardinal characteristics. In [8] we have shown that there are pairs of relations with no Borel morphism connecting them. The reason was a strong impact of the first of the two functions that constitute a morphism, the so-called function on the questions. In this work we investigate whether the second half, the function on the answers' side, has a similarly strong impact. The main question is: Does the nonexistence of a Borel morphism imply the non-existence of a morphism that is only Borel on the answers' side? We give sufficient conditions for an affirmative answer. The results are applied to the unsplitting relations where it has been open whether there is a morphism that is Borel on the answers' side.