Table of contents

The paper is concerned with properties of the modified P-integral and P-derivative, which are defined as multipliers with respect to the generalized Walsh-Fourier transform. Criteria for a function to have a representation as the P-integral or P-derivative of an L^p-function are given, and direct and inverse approximation theorems for P-differentiable functions are established. A relation between the approximation properties of a function and the behaviour of P-derivatives of the appropriate approximate identity is obtained. Analogues of Lizorkin and Taibleson's results on embeddings between the domain of definition of the P-derivative and Hölder-Besov classes are established. Some theorems on embeddings into BMO, Lipschitz and Morrey spaces are proved.

Bibliography: 40 titles.

We put forward a method for constructing semiorthogonal decompositions of the derived category of G-equivariant sheaves on a variety X under the assumption that the derived category of sheaves on X admits a semiorthogonal decomposition with components preserved by the action of the group G on X. This method is used to obtain semiorthogonal decompositions of equivariant derived categories for projective bundles and blow-ups with a smooth centre as well as for varieties with a full exceptional collection preserved by the group action. Our main technical tool is descent theory for derived categories.

Bibliography: 12 titles.

The paper is devoted to a new branch in the theory of one-dimensional variational problems with branching extremals, the investigation of one-dimensional minimal fillings introduced by the authors. On the one hand, this problem is a one-dimensional version of a generalization of Gromov's minimal fillings problem to the case of stratified manifolds. On the other hand, this problem is interesting in itself and also can be considered as a generalization of another classical problem, the Steiner problem on the construction of a shortest network connecting a given set of terminals. Besides the statement of the problem, we discuss several properties of the minimal fillings and state several conjectures.

Bibliography: 38 titles.

The functions on a space of dimension over the residue class ring  modulo  that are invariant with respect to the group form a commutative convolution algebra. We describe the structure of this algebra and find the eigenvectors and eigenvalues of the operators of multiplication by elements of this algebra. The results thus obtained are applied to solve the inverse problem for the hyperplane Radon transform on .

Bibliography: 2 titles.

Conditions for the existence of bounded and periodic solutions of the nonlinear functional differential equation

are presented, involving local linear approximations to the operator .

Bibliography: 23 titles.