Axial Borel functions

A function from the plane to the plane is axial if it does not change one coordinate. We show that every Borel permutation of the plane is a superposition of 11 Borel axial permutations.     

Towers of Borel functions

We give mathematical reformulations of the cardinals and in terms of families of Borel functions. As an application we show that is invariant under the addition of a single Cohen real.

     

Borel complexity of the space of probability measures

Using a technique developed by Louveau and Saint Raymond, we find the complexity of the space of probability measures in the Borel hierarchy: if is any non-Polish Borel subspace of a Polish space, then , the space of probability Borel measures on with the weak topology, is always true , where is the least ordinal such that is .

     

Borel extractions of converging sequences in compact sets of Borel functions

It is well known by a classical result of Bourgain–Fremlin–Talagrand that if K is a pointwise compact set of Borel functions on a Polish space then given any cluster point f of a sequence (f_n)_nω in K one can extract a subsequence (f_nk)_kω converging to f. In the present work we prove that this extraction can be achieved in a “Borel way.” This will prove in particular that the notion of analytic subspace of a separable Rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence.     

Measure-category properties of Borel plane sets and Borel functions of two variables

Let      

Canonical forms of pseudo-orthogonal matrices

Canonical forms are found for all real four-dimensional matrices of the pseudo-orthogonal group which differs from the Lorentz group only in that its metric has twoplus signs and twominus signs.     

Canonical contact forms on spherical CR manifolds

We construct the CR invariant canonical contact form can(J) on scalar positive spherical CR manifold (M,J), which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold ()/, where is a convex cocompact subgroup of Aut_CRS²ⁿ⁺¹=PU(n+1,1) and () is the discontinuity domain of . This contact form can be used to prove that ()/ is scalar positive (respectively, scalar negative, or scalar vanishing) if and only if the critical exponent ()<n (respectively, ()>n, or ()=n). This generalizes Nayatanis result for convex cocompact subgroups of SO(n+1,1). We also discuss the connected sum of spherical CR manifolds.     

Compact sets of continuity for Borel functions

We investigate connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set , where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming -determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented.     

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)     

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)     

Canonical forms for linear systems

This paper considers canonical forms for the similarity action of Gl(n) on $\text{∑}{}_{\mathrm{n,m}}\text{={(A,B)∈}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{n}}\text{×}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{m}}\text{}}$:

$\text{Gl(n×∑}{}_{\mathrm{n,m}}\text{→∑}{}_{\mathrm{n,m}}$

,

$\text{(H,(A,B))?(HAH}{}^{-1}\text{,HB)}$

Those canonical forms are obtained as an application of a more general method to select canonical elements M_c in the orbits $\text{O}{}_{\mathrm{M}}$ of a matrix group G acting on a set of matrices $\text{M}\text{?}\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$. We define a total order (?) on $\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$, different from the lexicographic order ^l? [0^l?x ? x <0, but $\text{0?x≠0 for x∈}\text{R}\text{]}$ and consider normalized $\text{O}{}_{\mathrm{M}}$-elements with a minimal number of parameters:

$\text{min{}\text{M}\text{?}\text{∈}\text{O}{}_{\mathrm{M}}\text{:}\text{M}\text{?}\text{normalized}}$

It is shown that the row and column echelon forms, the Jordan canonical form, and “nice” control canonical forms for reachable (A,B)-pairs have a homogeneous interpretation as such (?)-minimal orbit elements. Moreover new canonical forms for the general action (?) are determined via this method.     

Borel functions and separability of metric spaces

Let X and Y be metric spaces with X separable, and let \(f: X\rightarrow Y\) be a Borel function. Is then f(X) separable? In this paper, we prove that this problem is independent of ZFC. We also give a partial answer to an open problem which was asked by A. H. Stone.     

Canonical factorization of continuous functions on the -torus

In this article we study the canonical factorization of continuous complex-valued functions on the -dimensional torus belonging to a weighted Wiener algebra with respect to a linear order on the -tuples of integers. It is proved that a function has a canonical factorization in this algebra if and only if it has a logarithm belonging to this algebra. A second characterization is given in terms of winding numbers. Moreover, the maximal ideal spaces of the relevant Banach algebras are identified.