Table of contents

In this work two new invariants of virtual links are constructed: the even Alexander polynomial and the even quandle. The general idea behind the construction is to split the classical crossings into two types, the even and the odd ones, and then define different operations at the crossings of different types. On the other hand, the proposed construction is a realization of the same idea using two closely related languages: the language of quandles and the language of Alexander polynomials.

Bibliography: 15 titles.

We develop the Ercolani-Sinha construction of SU(2) monopoles for a five-parameter family of centred charge 3 monopoles. In particular we show how to solve the transcendental constraints arising on the spectral curve. For a class of symmetric curves the transcendental constraints become a number-theoretic problem and a recently proven identity of Ramanujan provides a solution.

Bibliography: 36 titles.

Let be a continuous function. It is shown that under certain assumptions on and weak solutions of the differential inequality on  are nonnegative. Some extensions of the result in the framework of subelliptic operators on Carnot groups are considered.

Bibliography: 19 titles.

For a skew product of interval maps with a closed set of periodic points, the dependence of the structure of its -limit sets on its differential properties is investigated. An example of a map in this class is constructed which has the maximal differentiability properties (within a certain subclass) with respect to the variable , is -smooth in the -variable and has one-dimensional -limit sets. Theorems are proved that give necessary conditions for one-dimensional -limit sets to exist. One of them is formulated in terms of the divergence of the series consisting of the values of a function of ; this function is the -norm of the deviation of the restrictions of the fibre maps to some nondegenerate closed interval from the identity on the same interval. Another theorem is formulated in terms of the properties of the partial derivative with respect to  of the fibre maps. A complete description is given of the -limit sets of certain class of -smooth skew products satisfying some natural conditions.

Bibliography: 33 titles.

Open, discrete -mappings in , , , are proved to be absolutely continuous on lines, to belong to the Sobolev class , to be differentiable almost everywhere and to have the -property (converse to the Luzin -property). It is shown that a family of open, discrete shell-based -mappings leaving out a subset of positive capacity is normal, provided that either has finite mean oscillation at each point or has only logarithmic singularities of order at most . Under the same assumptions on  it is proved that an isolated singularity of an open discrete shell-based -map is removable; moreover, the extended map is open and discrete. On the basis of these results analogues of the well-known Liouville, Sokhotskii-Weierstrass and Picard theorems are obtained.

Bibliography: 34 titles.