The notation and terminology used here have been introduced in the following papers: [17], [3], [4], [8], [13], [1], [2], [5], [15], [14], [21], [9], [12], [11], [16], [6], [20], [19], and [18]. For simplicity, we adopt the following rules: O1 is a non empty set, S1 is a σ-field of subsets of O1, P1 is a probability on S1, A is a sequence of subsets of S1, and n is an element of N. Let D be a s...

Recently, Weniger (delta sequence) method has been proposed by the authors of Ref. [1] for resummation of truncated perturbation series in quantum field theories. Those authors presented numerical evidence suggesting that this method works better than Padé approximants when we resum a function with singularities in the Borel plane but not on the positive axis. We present here numerical evidence...

— Colmez has given a recipe to associate a smooth modular representation Ω(W ) of the Borel subgroup of GL2(Qp) to a Fp-representation W of Gal(Qp/Qp) by using Fontaine’s theory of (φ,Γ)-modules. We compute Ω(W ) explicitly and we prove that if W is irreducible and dim(W ) = 2, then Ω(W ) is the restriction to the Borel subgroup of GL2(Qp) of the supersingular representation associated to W by ...

We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σ α -measurable functions (for every fixed 1 ď α ă ω1). Moreover, we present some results concerning those Borel functions which are ω-decomposable into continuous f...

— Colmez has given a recipe to associate a smooth modular representation Ω(W ) of the Borel subgroup of GL2(Qp) to a Fp-representation W of Gal(Qp/Qp) by using Fontaine’s theory of (φ,Γ)-modules. We compute Ω(W ) explicitly and we prove that if W is irreducible and dim(W ) = 2, then Ω(W ) is the restriction to the Borel subgroup of GL2(Qp) of the supersingular representation associated to W by ...

ω-powers of finitary languages are ω-languages in the form V , where V is a finitary language over a finite alphabet Σ. Since the set Σ of infinite words over Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers naturally arises and has been raised by Niwinski [Niw90], by Simonnet [Sim92], and by Staiger [Sta97b]. It has been proved in [Fin01]...

1. Suppose f is measurable. Then f−1({−∞}) ∈ M and f−1({∞}) ∈ M, because {−∞} and {∞} are Borel sets. If B ⊆ R is Borel then f−1(B) ∈M, and hence f−1(B) ∩ Y ∈M (since R is also Borel). Thus f is measurable on Y. Conversely, suppose that f−1({−∞}) ∈ M, f−1({∞}) ∈ M and f is measurable on Y. Let B ⊆ R be Borel. Then f−1(B) ∩ Y ∈ M, and f−1(B) = (f−1(B) ∩ Y ) ∪ (f−1(B) \ Y ). Clearly f−1(B) \ Y = ...

ω-powers of finitary languages are ω-languages in the form V , where V is a finitary language over a finite alphabet Σ. Since the set Σ of infinite words over Σ can be equipped with the usual Cantor topology, the question of the topological complexity of ω-powers naturally arises and has been raised by Niwinski [Niw90], by Simonnet [Sim92], and by Staiger [Sta97b]. It has been proved in [Fin01]...