Elements of Mathematics Functions of a Real Variable

Abstract

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On a conjecture by Pierre Cartier about a group of associators

“…The algebraic aspects of our regularization process based essentially on various products 1 among polyzêtas (see [33]) and its analytical aspects will be described, in Section 3.1, as the finite part, of the asymptotic expansions in different scales of comparison 2 [5]. It will be seen also, in Section 3.2, as the action of the differential Galois group of the polylogarithms 3 (recalled in Section 2.1.2)…”

“…In this way [39], polyzêtas do appear then as fundamental arithmetical constant for the asymptotic analysis and for the renormalization of the outputs and their successive derivations (Corollary 18) via the extended Fliess fundamental formula (Theorem 23). 5 Here, ǫ stands for the empty word over Y .…”

On spatial mappings with integral restrictions on the characteristic

“…In the present paper, we show that the mapping f extends by continuity to an isolated point of the boundary of D. Also we prove the equicontinuity of the families under consideration, assuming that f satisfies (4), where the function Q(x) satisfies (1), and the function Φ occurring in (1) satisfies condition (7). Also, we shall show that the restriction on the function Φ mentioned above, that is, property (7), is not only sufficient, but, in an appropriate sense, also necessary.…”

“…We use the symbol Γ to denote a fixed family of curves γ; by definition, f (Γ) = {f •γ|γ ∈ Γ}. The following definitions can be found, for example, in [22,Chapter 1,[1][2][3][4][5][6]. A Borel function ρ : R n → [0, ∞] is said to be admissible for a family Γ of curves γ in R n if the inequality γ ρ(x) |dx| ≥ 1 is true for the curvilinear integral of the first kind and for all curves γ ∈ Γ.…”

“…See, e.g., [3,10,11,12] and [20] for further details about inequalities of type (4). Note that, in the case where Q(x) ≤ K a.e., an expression like K · M (Γ(S 1 , S 2 , A)) arises on the right-hand side of inequality (4), see inequalities (4) and (3).…”