Related papers: Pad\'e interpolation for elliptic Painlev\'e equat…
A q-difference analogue of the fourth Painlev\'e equation is proposed. Its symmetry structure and some particular solutions are investigated.
In this article, we will report the recent developments on Lax pairs and Darboux transformations for Euler equations of inviscid fluids.
Isospectral problem of both 2D and 3D Euler equations of inviscid fluids, is investigated. Connections with the Clay problem are described. Spectral theorem of the Lax pair is studied.
Here we solve Pad\'e and Prony interpolation problems for the generalized exponential sums with equal weights: $$H_n(z; h)=\frac{\mu}{n}\sum_{k=1}^n h(\lambda_k z),\quad \text{where}\quad \mu,\lambda_k\in \mathbb{C},$$ and $h$ is a fixed…
Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.
Reducing the NP-problems to the convex/linear analysis on the Birkhoff polytope.
The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlev\'e equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the…
We prove doubling inequalities for solutions of elliptic systems with an iterated Laplacian as diagonal principal part and for solutions of the Lame' system of isotropic linearized elasticity. These inequalities depend on global properties…
The well known elliptic discrete Painlev\'e equation of Sakai is constructed by a standard translation on the $E_8^{(1)}$ lattice, given by nearest neighbor vectors. In this paper, we give a new elliptic discrete Painlev\'e equation…
In the space of all entire functions it is solved the problem of interpolation taking into account multiplicities by sums of the series of exponentials with the exponents from a given set. It is found a criterion of solubility of the…
First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and…
A completely integrable nonlinear partial differential equation (PDE) can be associated with a system of linear PDEs in an auxiliary function whose compatibility requires that the original PDE is satisfied. This associated system is called…
The paper concerns singular solutions of nonlinear elliptic equations.
A new Lax pair for the sixth Painlev\'e equation $P_{VI}$ is constructed in the framework of the loop algebra $\mathfrak{so}(8)[z,z^{-1}]$. The whole affine Weyl group symmetry of $P_{VI}$ is interpreted as gauge transformations of the…
In the space of holomorphic functions in a convex domain it is studied the interpolation problem by means of sums of the series of exponentials converging uniformly on all compact sets of the domain. The discrete set of the interpolation…
We give an explicit determinant formula for a class of rational solutions of a q-analogue of the Painlev\'e V equation. The entries of the determinant are given by the continuous q-Laguerre polynomials.
We present the bilinear forms of the (continuous) Painlev\'e equations obtained from the continuous limit of the analogous expresssions for the discrete ones. The advantage of this method is that it leads to very symmetrical results. A new…
We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -{\rm div~}(|\nabla…
We review recent results on the connection between Hermite-Pad\'e approximation problem, multiple orthogonal polynomials, and multidimensional Toda equations in continuous and discrete time. In order to motivate interest in the subject we…
We introduce a new approach to obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator being considered satisfies a certain type of weighted integral inequalities. The method is illustrated on…