We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris [1] and Mayer-Wolf [4]. We show that this relation is Borel bireducible to the universal relation for Borel actions of Polish groups.

We give a classical proof of the generalization of the characterization of smoothness to quotients of Polish spaces by Borel equivalence relations. As an application, we describe the extent to which any given Borel equivalence relation on a Polish space is encoded by the corresponding σ-ideal generated by the family of Borel sets on which it is smooth.

1 Probability measures on metric spaces 1 1.1 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Borel probability measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Narrow convergence of measures . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 The bounded Lipschitz metric . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5...

In 1936, Alan Turing wrote a remarkable paper giving a negative answer to Hilbert’s Entscheidungsproblem [29]. Restated with modern terminology and in its relativized form, Turing showed that given any infinite binary sequence x ∈ 2ω, the set x′ of Turing machines that halt relative to x is not computable from x. This function x 7→ x′ is now known as the Turing jump, and it has played a singula...

The singular part of Borel transform of a QCD amplitude near the infrared renormalon can be expanded in terms of higher order Wilson coefficients of the operators associated with the renormalon. In this paper we observe that this expansion gives nontrivial constraints on the Borel amplitude that can be used to improve the accuracy of the ordinary perturbative expansion of the Borel amplitude. I...

The singular part of Borel transform of a QCD amplitude near the infrared renormalon can be expanded in terms of higher order Wilson coefficients of the operators associated with the renormalon. In this paper we observe that this expansion gives nontrivial constraints on the Borel amplitude that can be used to improve the accuracy of the ordinary perturbative expansion of the Borel amplitude. I...

We obtain some results about Borel maps with meager fibers on completely metrizable separable spaces. The results are related to a recent dichotomy by Sabok and Zapletal, concerning Borel maps and σ-ideals generated by closed sets. In particular, we give a “classical” proof of this dichotomy. We shall also show that for certain natural σ-ideals I generated by closed sets in compact metrizable s...

We first consider infinite two-player games on pushdown graphs. In previous work, Cachat, Duparc and Thomas [4] have presented a winning decidable condition that is Σ3-complete in the Borel hierarchy. This was the first example of a decidable winning condition of such Borel complexity. We extend this result by giving a family of decidable winning conditions of arbitrary finite Borel complexity....

We call a subset S of a topological vector space V linearly Borel, if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V . It will be shown that a Hamel base of an infinite dimensional Banach space can never be linearly Borel. This answers a question of Anatolij Plichko. In the sequel, let X be any infinite dimensional Banach space. A subset S ...

1.1. Brief overview. The theory of partial differential equations when one, or more variables, is in the complex domain, and approaches a characteristic variety has only recently started to develop. In their paper [11], generalized in [12], O. Costin and S. Tanveer proved existence and uniqueness of solutions with given initial conditions, for quasilinear systems of evolution equations in a lar...