Table of contents

We consider surfaces whose points are the lines on the real three-dimensional varieties of degree 3. These surfaces are called Fano surfaces. This paper deals with finding the topological types, that is, a topological classification, of real Fano surfaces. Moreover, we prove that the equivariant topological type of the corresponding complex Fano surface with the involution of complex conjugation determines the rigid isotopy class of the corresponding real three-dimensional cubic.

We obtain upper and lower bounds for the best accuracy of approximation in Stechkin's problem for the differentiation operator and in the problem of the reconstruction of the derivative from the values of the function at a given number of points for Nikol'skii and Besov classes of functions satisfying mixed Hölder's conditions. These estimates give the order of these quantities for almost all values of the parameters involved.

We establish conditions guaranteeing that a group possesses the following property: there is a number such that if elements , of generate a finite subgroup then lies in the normalizer of . These conditions are of a quite special form. They hold for groups with relations of the form which appear as approximating groups for the free Burnside groups of sufficiently large even exponent . We extract an algebraic assertion which plays an important role in all known approaches to substantial results on the groups of large even exponent, in particular, to proving their infiniteness. The main theorem asserts that when is divisible by 16, has the above property with .

The Khovanov homology theory over an arbitrary coefficient ring is extended to the case of virtual knots. We introduce a complex which is well-defined in the virtual case and is homotopy equivalent to the original Khovanov complex in the classical case. Unlike Khovanov's original construction, our definition of the complex does not use any additional prescription of signs to the edges of a cube. Moreover, our method enables us to construct a Khovanov homology theory for `twisted virtual knots' in the sense of Bourgoin and Viro (including knots in three-dimensional projective space). We generalize a number of results of Khovanov homology theory (the Wehrli complex, minimality problems, Frobenius extensions) to virtual knots with non-orientable atoms.

We study problems of approximation of functions on  in the metric of  with power weight using generalized Bessel shifts. We prove analogues of direct Jackson theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Bessel shifts. We establish the equivalence of the modulus of smoothness and the -functional. We define function spaces of Nikol'skii-Besov type and describe them in terms of best approximations. As a tool for approximation, we use a certain class of entire functions of exponential type. In this class, we prove analogues of Bernstein's inequality and others for the Bessel differential operator and its fractional powers. The main tool we use to solve these problems is Bessel harmonic analysis.

We establish formulae for isometric embeddings and immersions of Möbius bands with a locally Euclidean metric and study extrinsic geometric properties of these surfaces. We consider both standard Möbius bands corresponding to embeddings of a rectangular Möbius strip and general Möbius bands, in particular, those with generators orthogonal to the directrix.