Related papers: Th\'eor\`emes de Borel avec contraintes

The following extension of Bohr's theorem is established: If a somewhere convergent Dirichlet series $f$ has an analytic continuation to the half-plane $\mathbb{C}_\theta = \{s = \sigma+it\,:\, \sigma>\theta\}$ that maps $\mathbb{C}_\theta$…

As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\in {\mathbb R}$ is the value of Euler's gamma function $\Gamma(1+x)$. In this connection, the notion of a binomial coefficient naturally…

This note presents a general theorem about the cohomology of finite dimensional Lie algebras of arbitrary characteristic. As an application we compute the cohomology of the Borel subalgebra of sl(N).

We study asymptotics of integrals of certain rational functions that depend on parameters in a field $K$ of characteristic zero. We use formal power series to represent the integral and prove certain identities about its coefficients…

We define and study expansion problems on countable structures in the setting of descriptive combinatorics. We consider both expansions on countable Borel equivalence relations and on countable groups, in the Borel, measure and category…

For a given length and a given degree and an arbitrary partition of the positive integers, there always is a cell containing a polynomial progression of that length and that degree; moreover, the coefficients of the generating polynomial…

A classical theorem of d'Alembert states that if a polynomial P(x) with real coefficients has a non-real root x=a+ib, then it also has a root x=a-ib. We give a short and elementary inductive proof that avoids any properties of the complex…

A family of formal power series, such that its coefficients satisfy a recursion formula, is characterized in terms of the summability, in the sense of J. P. Ramis, of its elements along certain well chosen directions. We describe a set of…

We consider the theory of algebraically closed fields of characteristic zero with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact the theory of equations in exponential sums. In an earlier paper we have described…

Approximation theory has long been concerned with the development of positive linear operators that effectively approximate classes of functions. Among the most well-known results in this area are Korovkin-type approximation theorems, which…

Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular,…

Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes,…

We discuss certain aspects of the formal calculus used to describe vertex algebras. In the standard literature on formal calculus, the expression $(x+y)^{n}$, where $n$ is not necessarily a nonnegative integer, is defined as the formal…

We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an $\omega$-Erdos cardinal, we determine which of these theories are Borel complete. We develop machinery,…

Many theorems in complex analysis propagate analyticity, such as the Forelli theorem, edge-of-the-wedge theorem and so on. We give a germination theorem which allows for general analytic propagation in complete normed fields. In turn, we…

Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most…

We extend the recently introduced setting of coherent differentiation for taking into account not only differentiation, but also Taylor expansion in categories which are not necessarily (left)additive. The main idea consists in extending…

We generalize the Pierce representation theorem for (commutative) rings with unit to other algebraic categories with Definable Factor Congruences by using tools from topos theory. Of independent interest, we prove that an algebraic category…

In this paper we prove the Bohr Theorem for slice regular functions. Following the historical path that led to the proof of the classical Bohr Theorem, we also extend the Borel-Carath\'eodory Theorem to the new setting.

A definition of summability is put forward in the framework of general Carleman ultraholomorphic classes in sectors, so generalizing $k-$summability theory as developed by J.-P. Ramis. Departing from a strongly regular sequence of positive…