Related papers: La formule de Cauchy dans les alg\`ebres
We define and study a generalization of the analytic Cauchy problem, that specializes to the Cauchy-Kowaleskaya-Kashiwara problem in the linear case. The main leitmotive of this text is to adapt Kashiwara's formulation of this problem both…
We introduce the multivariate analogue of the well known inequality $1+x\leq \mathrm{e}^x$, for an abstract non negative real number $x$. The result emerges from the study of the blow up time of certain solutions of the Cauchy problem for a…
We generalize the results in [2] giving a reduction algorithm allowing to compute the index of seaweed subalgebras of classical simple Lie algebras. We thus are able to obtain the index of some interesting families of seaweed subalgebras…
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding…
An introductory review of algebraic classification of the Weyl tensor and algebraically special solutions in higher dimensions.
In this work of divulgation, the global behavior of the Lyness equation is described. Results are obtained for different values of the parameter $\alpha$.
The well known Duhamel's principle allows to reduce the Cauchy problem for linear inhomogeneous partial differential equations to the Cauchy problem for the corresponding homogeneous equation. In the paper one of the possible…
We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-L\'evy type theorem and a factorization theorem. We give applications to Toeplitz and…
The operators of fractional calculus come in many different types, which can be categorised into general classes according to their nature and properties. We conduct a formal study of the class known as weighted fractional calculus and its…
The purpose of this note is to compare various approximation methods as applied to the inverse of the Bessel function of the first kind, in a given domain of the complex plane.
Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are…
In this paper, we estimate the Hilbert-Kunz multiplicity of the (extended) Rees algebras in terms of some invariants of the base ring. Also, we give an explicit formula for the Hilbert-Kunz multiplicities of Rees algebras over Veronese…
We consider the Cauchy problem for nonlinear Schr\"odinger equations in a general domain $\Omega\subset\mathbb{R}^N$. Construction of solutions has been only done by classical compactness method in previous results. Here, we construct…
In these informal lecture notes we outline different approaches used in doing calculations involving the Dirac equation in curved spacetime. We have tried to clarify the subject by carefully pointing out the various conventions used and by…
The Lerche-Newberger formula simplifies harmonic sums of Bessel functions and has seen application in plasma physics and frequency modulated quantum systems. In this paper, we rigorously prove the formula and extend the classical result to…
Starting from a description of various generalized function algebras based on sequence spaces, we develop the general framework for considering linear problems with singular coefficients or non linear problems. Therefore, we prove…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
The concept of derivation for Lie-Yamaguti algebras is generalized in this paper. A quasi-derivation of an LY-algebra is embedded as derivation in a larger LY-algebra. The relationship between quasi-derivations and robustness of…
We treat the classical notion of convexity in the context of hard real analysis. Definitions of the concept are given in terms of defining functions and quadratic forms, and characterizations are provided of different concrete notions of…
We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary…