A class of multidimensional integrable hierarchies connected with commutation of general (unreduced) (N+1)-dimensional vector fields containing derivative over spectral variable is considered. They are represented in the form of generating equation, as well as in the Lax-Sato form. A dressing scheme based on nonlinear vector Riemann problem is presented for this class. The hierarchies connected...

As a basic model, the fourth-order quasilinear thin film equation (the TFE– 4) with a concentration-dependent doubly degenerate mobility coefficient vanishing at two equilibrium levels u = ±1, ut = −(|1− u|uxxx)x , is studied. The basic Riemann problem for the TFE–4 with discontinuous initial data

We deal with initial-boundary value problems for systems of conservation laws in one space dimension and we focus on the so-called boundary Riemann problem. It is known that, in general, different viscous approximations provide different limits. We establish sufficient conditions to conclude that two different approximations lead to the same limit. As an application of this result, we show that...

Fix positive integers a and b such that a > b ≥ 2 and a positive real δ > 0. Let S be a planar set of diameter δ having the following property: for every a points in S, at least b of them have pairwise distances that are all less than or equal to 2. What is the maximum Lebesgue measure of S? In this paper we investigate this problem. We discuss the, devious, motivation that leads to its formula...

We establish estimates for the number of ways to represent any reduced residue class as a product prime and an integer free small factors. Our best results is conditional on Generalised Riemann hypothesis (GRH). As corollary, we make progress conjecture Erdös, Odlyzko, Sárközy.

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. generalisation that we replace exponential function with Mittag-Leffler-type function, oscillatory integrals appearing in analysis time-fractional partial differential equations. More specifically, integral form I?,?(?)=?RE?,?(i???(x))?(x)dx, for range 00. This extends variety estimat...