Table of contents

For the problem , , , one shows the existence and uniqueness of a solution obtainable as a limit as tends to zero of the bounded self-similar solutions of the regularized problem with additional viscosity term , , in the second equation. The structure of the solutions is described in detail, in particular, when they contain vacuum states.

A new class of extrapolation functors on the scale of -spaces is introduced, allowing one to take for its "limiting spaces" two symmetric spaces "close" to and . Crucial here are the extrapolation relations for the Peetre - and -functionals for the Banach couples and , respectively and , , are Zygmund spaces). The real method of operator interpolation is used.

Set , where is an irrational number, and let be the radius of holomorphy of the Rogers-Ramanujan function

As is known, and for each there exists such that . It is proved here that the function is meromorphic not only in the disc , but also in the disc , which is larger for ; and that the Rogers-Ramanujan continued fraction converges to on compact subsets contained in , where is the union of circles with centres at and passing through the poles of . The convergence of the Rogers-Ramanujan continued fraction in the domain was established earlier by Lubinsky.

The -differentiability in the topology of the Sobolev space of weakly contact maps of Carnot groups is proved. The -differentiability in the sense of Pansu of contact maps in the class , , and other results are established as consequences. The method of proof is new even in the case of a Euclidean space and yields, for instance, a new proof of well-known results of Reshetnyak and Calderon-Zygmund on the differentiability of functions of Sobolev classes. In addition, a new proof of Lusin's condition is given for quasimonotone maps in the class . As a consequence, change-of-variables formulae are obtained for maps of Carnot groups.

What general regularity manifests itself in the fact that a triangle, and in general any convex polygon, cannot be tessellated by non-convex quadrangles? Another question: it is known that for the plane cannot be tessellated by convex -gons if their diameters are bounded, while the areas are separated from zero; can this fact be generalized for non-convex polygons? In the present paper we introduce the characteristic of a polygon . We answer the above questions in terms of and then study tessellations of the plane by -gons equivalent to , that is, with the same sequence of angles greater than and smaller than .

A formal Schrödinger operator of the form

in is considered, where is a bounded measurable vector-valued function and both and are measures satisfying certain additional conditions. It is shown that one can give meaning to such an operator as a lower bounded self-adjoint operator in . The corresponding heat kernel is constructed and its small-time asymptotics are obtained. A rigorous Feynman path integral representation for the solutions of the heat and Schrödinger's equations with generator is given.

In the one-dimensional case, for a function satisfying the Gurov-Reshetnyak condition, the infimum of the indices of the Muckenhoupt classes containing this function is found. It is also shown that each Muckenhoupt class lies in some Gurov-Reshetnyak class.

Exact expressions are obtained for the distribution of the total number of crossings of a strip by sample paths of a random walk whose jumps have a two-sided geometric distribution. The distribution of the number of crossings during a finite time interval is found in explicit form for walks with jumps taking the values . A limit theorem is proved for the joint distribution of the number of crossings of an expanding strip on a finite (increasing) time interval and the position of the walk at the end of this interval, and the corresponding limit distribution is found.

For a broad class of matrices (discrete analogues of typical integral operators) their approximability by a sum of direct products of matrices of smaller size is demonstrated. Estimates of the number of terms (the tensor rank) and the corresponding error are obtained. It is shown that, as a method of data compression, tensor approximations provide superlinear compression.