Related papers: Rules of discretization for Painlev\'e equations
Conditional specification of distributions is a developing area with increasing applications. In the finite discrete case, a variety of compatible conditions can be derived. In this paper, we propose an alternative approach to study the…
We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference…
In this paper, two methods are employed to investigate for which values of the parameters, if any, the two-dimensional real Landau-Ginzburg equation possesses the Painleve property. For an ordinary differential equation to have the Painleve…
In materials science, wedge disclinations are defects caused by angular mismatches in the crystallographic lattice. To describe such disclinations, we introduce an atomistic model in planar domains. This model is given by a…
We introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving. The method is discretization-independent provided elementary symmetry and skew-symmetry properties are satisfied by…
From the distributional characterizations that lie at the heart of Stein's method we derive explicit formulae for the mass functions of discrete probability laws that identify those distributions. These identities are applied to develop…
Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
In this paper some open problems for Painlev\'e equations are discussed. In particular the following open problems are described: (i) the Painlev\'e equivalence problem; (ii) notation for solutions of the Painlev\'e equations; (iii)…
Discrete Painlev\'e equations constitute a famous class of integrable non-autonomous second order difference equations. A classification scheme proposed by Sakai interprets a discrete Painlev\'e equation as a birational map between…
The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such…
We apply Painlev\'e test to the most general variable coefficient nonlinear Schrodinger (VCNLS) equations as an attempt to identify integrable classes and compare our results versus those obtained by the use of other tools like…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
The problem of Painleve classification of ordinary differential equations lasting since the end of XIX century saw significant advances for the limited equation order, however not that much for the equations of higher orders. In this work…
Based on the convex least-squares estimator, we propose two different procedures for testing convexity of a probability mass function supported on N with an unknown finite support. The procedures are shown to be asymptotically calibrated.
The derivation of second-order ordinary differential equations (ODEs) as continuous-time limits of optimization algorithms has been shown to be an effective tool for the analysis of these algorithms. Additionally, discretizing…
In this paper we explore the discretization of Euler-Poincar\'e-Suslov equations on $SO(3)$, i.e. of the Suslov problem. We show that the consistency order corresponding to the unreduced and reduced setups, when the discrete reconstruction…
We extend the Gibbs conditioning principle to an abstract setting combining infinitely many linear equality constraints and non-linear inequality constraints, which need not be convex. A conditional large large deviation principle (LDP) is…
Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of…
The oscillatory behavior of the solutions to a differential equation with several non-monotone delay arguments and non-negative coefficients is studied. A new sufficient oscillation condition, involving lim sup, is obtained. An example…