Related papers: Linear differential equations and Hurwitz series
Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2,…
Below, the explicit solution to a certain finite-difference equation is given and the required steps for derivation of these results are outlined. Everything is included as Mathematica formulae, so the notebook itself can be used for…
We prove that almost all solutions of the Markoff-Hurwitz equation over a residue field modulo $p$ can be obtained from one another by a chain of natural transformations. We also study recurrence sequences considered modulo prime $p$.
We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on…
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…
We introduce a natural structure of a semigroup (isomorphic to a factorization semigroup of the unity in the symmetric group) on the set of irreducible components of Hurwitz space of marked degree $d$ coverings of $\mathbb P^1$ of fixed…
In this paper, we develop a representation-theoretic formulation of discrete-time linear systems. We show that such systems are naturally viewed as representations of time groups acting on vector spaces, thereby endowing the state space…
In this paper, we prove the partial linearization for n-dimensional nonautonomous differential equations. The conditions are formulated in terms of the dichotomy spectrum.
We adapt the method of solution regions to prove new existence and localization results for systems of discontinuous differential equations. Some assumptions concerning the definition of a solution region are relaxed and thus our results…
A full Lie point symmetry analysis of rational difference equations is performed. Non-trivial symmetries are derived and exact solutions using these symmetries are obtained.
We give a complete solution of the linearization problem in the plane Cremona group over an algebraically closed field of characteristic zero.
It is well recognized that new types of exact travelling wave solutions to nonlinear partial differential equations can be obtained by modifications of the methods which are in hand. In this study, we extend the class of auxiliary equations…
We apply results proved in [Li19] to the linear order expansions of non-trivial free homogeneous structures and the universal n-linear order for $n\geq 2$, and prove the simplicity of their automorphism groups.
We study linear ordinary differential equations which are analytically parametrized on Hermitian symmetric spaces and invariant under the action of symplectic groups. They are generalizations of the classical Lam\'e equation. Our main…
The questions of global topological, smooth and holomorphic classifications of the differential systems, defined by covering foliations, are considered. The received results are applied to nonautonomous linear differential systems and…
Consider a third order linear differential equation $L(f)=0$, where $L\in\mathbb{Q}(z)[\partial_z]$. We design an algorithm computing the Liouvillian solutions of $L(f)=0$. The reducible cases devolve to the classical case of second order…
In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this…
We study the irreducible components of special loci of curves whose group of symmetries is given as certain group extension. We introduce some relative Hurwitz data, which we show by using mixed \'etale cohomology theory, identifies some…
We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…