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 paper addresses generalized Borel summability of “1+” difference equations in “critical time”. We show that the Borel transform Y of a prototypical such equation is analytic and exponentially bounded for <(p) < 1 but there is no analytic continuation from 0 toward +∞: the vertical line ` := {p : <(p) = 1} is a singularity barrier of Y . There is a unique natural continuation through the bar...

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall’s condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theore...

The report is organized as follows: section 2 presents a description of texture mapping and a justification for using a cache dedicated to texturing. Section 3 describes the basic texture caching architecture. The rest of the report review the successive improvements to this basic architecture found in the literature: multilevel caching (section 4), prefetching (section 5), parallel texture cac...

We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of Polish groups: universal actions, invariant probability measures, and the Topological Vaught Conjecture. We establish the existence of universal actions for any given Polish ...

A Borel equivalence relation on a Polish space is countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence rel...

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