相关论文: The rational generalized integrating factors for f…
We study conditions under which a partial differential operator of arbitrary order $n$ in two variables or ordinary linear differential operator admits a factorization with a first-order factor on the left. The factorization process…
An extension of the ideas of the Prelle-Singer procedure to second order differential equations is proposed. As in the original PS procedure, this version of our method deals with differential equations of the form…
Let k be a number field. It is well known that the set of sequences composed by Taylor coefficients of rational functions over k is closed under component-wise operations, and so it can be equipped with a ring structure. A conjecture due to…
The explicit solution of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODE s with second-degree polynomial right-hand sides, hence featuring 12 a prior arbitrary (time-independent)…
We have already dealt with the problem of solving First Order Differential Equations (1ODEs) presenting elementary functions before in [1, 2]. In this present paper, we have established solid theoretical basis through a relation between the…
In this article we present logarithmic methods for solving first order and second order ordinary differential equations. The essence of the method is that we apply the basic properties derivatives and logarithms to reduce the number of…
For a large class of systems of o.d.e.'s which have first integrals, the method of arrays yields the following results: i) The first integrals $I$ can be found by solving systems of linear equations. ii) How the first integral $I$ responds…
Valuation algebras abstract a large number of formalisms for automated reasoning and enable the definition of generic inference procedures. Many of these formalisms provide some notions of solutions. Typical examples are satisfying…
In this paper rational solutions of the fifth Painlev\'e equation are discussed. There are two classes of rational solutions of the fifth Painlev\'e equation, one expressed in terms of the generalised Laguerre polynomials, which are the…
Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs…
In this short communication we introduce a rather simple autonomous system of 2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose initial-values problem is explicitly solvable by algebraic operations. Its ODEs…
The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by…
We study the general rational solution of the Yang-Baxter equation with the symmetry algebra sl(3). The R-matrix acting in the tensor product of two arbitrary representations of the symmetry algebra can be represented as the product of the…
Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time…
In this paper, we introduce the notion of generalized rational Okamoto-Painlev\'e pair (S, Y) by generalizing the notion of the spaces of initial conditions of Painlev\'e equations. After classifying those pairs, we will establish an…
We present a general algorithm for constructing a free resolution for unit groups of orders in semisimple rational algebras. The approach is based on computing a contractible $G$-complex employing the theory of minimal classes of quadratic…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$…
In the present paper we derive complicated families of orthogonal polynomials in one variable from scratch using the known ones as building blocks. We recall the basics of operational formalism and introduce the notations we use throughout…