Table of contents

In this paper, we consider stochastic Schrödinger equations with two-dimensional white noise. Such equations are used to describe the evolution of an open quantum system undergoing a process of continuous measurement. Representations are obtained for solutions of such equations using a generalization to the stochastic case of the classical construction of Feynman path integrals over trajectories in the phase space.

We prove that every function  of class  subholomorphic in  can be extended to a subholomorphic function of class  in the whole with an estimate for the -norm, where and  is an arbitrary Jordan -domain in . We obtain some corollaries and an analogue of the above assertion for the classes with .

Research in convex analysis (in particular, in the theory of strongly convex sets developed in recent years) has made it possible to obtain important results in approximation theory, the theory of extremal problems, optimal control and differential game theory [1]-[3]. In many problems there arise non-convex sets that have weakened convexity properties, which enables one to study them using the methods of convex analysis. In this paper we study new properties of sets that are weakly convex in the sense of Vial or Efimov-Stechkin, that is, in the direct and dual senses. We establish relations between these two concepts of weak convexity. For subsets of Hilbert space that are weakly convex in the sense of Vial we prove a theorem on relative connectedness and a support principle.

We compute the topological type of the real part of the Fano surface that parametrizes the set of real lines a non-singular real M-threefold. When studying Fano surfaces, we use the results and constructions in [3] on the intermediate Jacobian of a three-dimensional complex cubic. We begin by computing the topological type of the real part of the Fano surface that parametrizes the set of real lines on a singular real M-cubic with a single simple singular point.

We study graphs in which for every edge and all -subgraphs are 2-cocliques. We give a description of connected edge-regular graphs for . In particular, the following examples confirm that the inequality is a sharp bound for strong regularity: the -gon, the icosahedron graph, the graph in and the distance-regular graph of diameter 4 with intersection massive , which is an antipodal 3-covering of the strongly regular graph with parameters .

It is shown that two strictly pseudoconvex Stein domains with real-analytic boundaries have biholomorphic universal coverings provided that their boundaries are locally biholomorphically equivalent. This statement can be regarded as a higher-dimensional analogue of the uniformization theorem.

It is shown that if two strictly pseudoconvex Stein domains with real-analytic boundaries have biholomorphic universal coverings, then their boundaries are locally biholomorphically equivalent.

Given and any (Jordan) -domain  in , we prove that any function of class  that is subharmonic in  can be extended to a function of class  that is subharmonic on the whole and give an estimate of the -norm of its gradient. The corresponding assertion for is false even for discs. These results also hold for balls  in , . We also obtain some corollaries, including the corresponding assertions on the -extension of subharmonic functions.

We prove the birational superrigidity of direct products of primitive Fano varieties of the following two types: either is a general hypersurface of degree , , or is a general double space of index 1, . In particular, every structure of a rationally connected fibre space on  is given by the projection onto a direct factor. The proof is based on the connectedness principle of Shokurov and Kollár and the technique of hypertangent divisors.

In 1998 the first author announced a theorem stating that every primitive -dimensional parallelohedron can be represented, up to an affine transformation, as a weighted Minkowski sum of parallelohedra belonging to a certain finite set of -dimensional mainstay parallelohedra situated in a special way. This paper contains a detailed proof of this theorem in a refined and definitive form.

We use local field theory to study a special class of discrete dynamical systems, where the function being iterated is a polynomial whose coefficients belong to the ring of integers in a -adic field.