Andrei Sobolevski

This paper investigates algorithms for solving discrete stationary (or) matrix Bellman equations over semirings, in particular over tropical and idempotent semirings, Also there are presented some original algorithms, applications and programmed realization.

Characteristic curves of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth "viscosity" solutions, which give rise to discontinuous velocity fields, this picture holds only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we discuss two physically meaningful regularisation procedures, one corresponding to vanishing viscosity and another to weak noise limit. We show that for any convex Hamiltonian, a viscous regularization allows to construct a nonsmooth flow that extends particle trajectories and determines dynamics inside the shock manifolds. This flow consists of integral curves of a particular "effective" velocity field, which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds. The effective velocity field arising in the weak noise limit is generally non-unique and different from the viscous one, but in both cases there is a fundamental self-consistency condition constraining the dynamics.

We consider a variational problem in a planar convex domain, motivated by statistical mechanics of crystal growth in a saturated solution. The minimizers are constructed explicitly and are completely characterized

We consider a variational problem in a planar convex domain, motivated by statistical mechanics of crystal growth in a saturated solution. The minimizers are constructed explicitly and are completely characterized.

We propose a toy model of a heteropolymer chain capable of forming planar secondary structures typical for RNA molecules. In this model, the sequential intervals between neighboring monomers along a chain are considered as quenched random variables, and energies of nonlocal bonds are assumed to be concave functions of those intervals. A few factors are neglected: the contribution of loop factors to the partition function, the variation in energies of different types of complementary nucleotides, the stacking interactions, and constraints on the minimal size of loops. However, the model captures well the formation of folded structures without pseudoknots in an arbitrary sequence of nucleotides. Using the optimization procedure for a special class of concave-type potentials, borrowed from optimal transport analysis, we derive the local difference equation for the ground state free energy of the chain with the planar (RNA-like) architecture of paired links. We consider various distribution functions of intervals between neighboring monomers (truncated Gaussian and scale free) and demonstrate the existence of a topological crossover from sequential to essentially nested configurations of paired links.