Alexander Vasil’ev

Dmitri Prokhorov; Alexander Vasil’ev

We consider the Lowner differential equation generating univalent maps of the unit disk (or of the upper half-plane) onto itself minus a single slit. We prove that the circular slits, tangent to the real axis are generated by Holder continuous driving terms with exponent 1/3 in the Lowner equation. Singular solutions are described, and the critical value of the norm of driving terms generating quasisymmetric slits in the disk is obtained.

Der-Chen Chang; Irina Markina; Alexander Vasil’ev

Abstract Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of three-dimensional sphere and some partial results are obtained in the general case. The Carnot–Caratheodory distance is calculated. We also present some motivations related to quantum mechanics.

Alexander Vasil’ev

The main goal of the paper is to apply methods of the theory of univalent functions to some problems of fluid mechanics. Our interest centers on free boundary problems. We study the time evolution of the free boundary of a viscous fluid in the zero- and non-zero-surface-tension models for planar flows in Hele-Shaw cells either with an extending to infinity free boundary or with a bounded free boundary. We consider special classes of univalent functions which admit an explicit geometric interpretation to characterize the shape of the free interface. The paper contains a survey part as well as new results. We also set up some new problems.

Filippo Bracci; Manuel D. Contreras; Santiago Díaz-Madrigal; Alexander Vasil’ev

In this paper we present a historical and scientific account of the development of the theory of the Lowner–Kufarev classical and stochastic equations spanning the 90-year period from the seminal paper by K. Lowner in 1923 to recent generalizations and stochastic versions and their relations to conformal field theory.

Georgy Ivanov; Alexander Vasil’ev

We consider evolution in the unit disk in which the sample paths are represented by the trajectories of points evolving randomly under the generalized Loewner equation. The driving mechanism differs from the SLE evolution, but nevertheless solutions possess similar invariance properties.

Mauricio Godoy Molina; Boris Kruglikov; Irina Markina; Alexander Vasil’ev

In the present paper, we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions for each of the choices. Explicit criteria for rigidity of pseudo H- and J-type algebras are given. In particular, we establish the relation of the so-called

Melkana A. Brakalova; Irina Markina; Alexander Vasil’ev

Georgy Ivanov; Nam-Gyu Kang; Alexander Vasil’ev

J^2

Alexey Tochin; Alexander Vasil’ev

Anastasia Frolova; Dmitry Khavinson; Alexander Vasil’ev

J2-condition to rigidity, and we explore these conditions in relation to pseudo H-type algebras.