The perturbative expansions in weak coupling constant in quantum field theories are in general divergent, with the coefficients growing in factorial. One way to make a rigorous sense out of the divergent series is to resum the series using Borel transform. When the large order behavior of the expansion is sign-alternating Borel resummation may yield the full amplitude. But in QCD the series are...

of the absolute Galois group of a number field F . Assume that ρ̄ is ordinary in the sense that the image of any decomposition group at a place v dividing p lies in some Borel subgroup Bv of G. Assume also that ρ̄ satisfies the conditions of [11, Section 7] which guarantee that it has a reasonable deformation theory; see Section 3.1 for details. In this paper we show that the Iwasawa invariants o...

We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures. Introduction. The Borel reducibility theory of Pol...

— KURATOWSKI showed that the derived set operator D, acting on the space of closed subsets of the Cantor space 2 ,̂ is a Borel map of class exactly two and posed the problem of determining the precise classes of the higher order derivatives D". In part I of our work [Bull. Soc. Math. France, 110, 4, 1982, p. 357-380], we obtained upper and lower bounds for the Borel class of Z>" and in particula...

It is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be Borel summable to the solutions themselves appear also to be the unique solutions to the 1D Schrödinger equation having this property. Namely, it is shown in this paper that for the polynomial potentials the Borel function defined by the fundamental solutions can be conside...

We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain