Related papers: The Peano-Baker series
In this paper we develop a new method for numerically approximating sensitivities in parameter-dependent ordinary differential equations (ODEs). Our approach, intended for situations where the standard forward and adjoint sensitivity…
Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the…
We present an alternative cyclic proof system for Peano arithmetic that could be simpler than the existing ones and well-adapted both for proof analysis and for automatizing inductive proof search. In addition, we will show how various…
Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expensive to compute. In this paper, we derive analytic…
We propose a framework to define solutions of ODE systems under a novel condition that goes well beyond the usual continuity condition required in the classical theory of ODEs (Peano's or Picard's theorems). We illustrate our results with…
Many important systems across biology, engineering, physics, and economics are characterized by polynomial ordinary differential equations (ODEs), yet analytical solutions are rare. We develop a framework for identifying and solving a broad…
This note presents a discussion of the algebraic and combinatorial aspects of the theory of pure O-sequences. Various instances where pure O-sequences appear are described. Several open problems that deserve further investigation are also…
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and…
We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the…
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued…
This papers studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and provides several implicit characterizations…
We propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.
In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of…
We present new proofs to four versions of Peano's Existence Theorem for ordinary differential equations and systems. We hope to have gained readability with respect to other usual proofs. We also intend to highlight some ideas due to Peano…
The solution of systems of non-autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in…
Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees,…
We classify the possible Scott complexities for models of Peano arithmetic. We construct models of particular complexities by first giving a complete Scott analysis of colored linear orderings and constructing models of Peano arithmetic…
The ``Painlev\'e analysis'' is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory…