Sergey P. Tsarev

Hyperdeterminants as integrable discrete systems

We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A Cayley in 1845. We prove integrability (understood as 4D consistency) of a nonlinear difference equation defined by the 2 × 2 × 2-hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants of any size are integrable. We show that this hypothesis already fails in the case of the 2 × 2 × 2 × 2-hyperdeterminant.

Differential transformations of parabolic second-order operators in the plane

Darboux’s classical results about transformations of second-order hyperbolic equations by means of differential substitutions are extended to the case of parabolic equations of the form Lu = (Dx2 + a(x, y)Dx + b(x, y)Dy + c(x, y))u = 0. We prove a general theorem that provides a way to determine admissible differential substitutions for such parabolic equations. It turns out that higher order transforming operators can always be represented as a composition of first-order operators that define a series of consecutive transformations. The existence of inverse transformations imposes some differential constrains on the coefficients of the initial operator. We show that these constraints may imply famous integrable equations, in particular, the Boussinesq equation.

On Local Description of Two-Dimensional Geodesic Flows with a Polynomial First Integral

In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows us to present infinitely many particular metrics by the generalized hodograph method.

The curvature line parametrization from circular nets on a surface

We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets (circular nets and planar quadrilateral nets, respectively) with edges of order ϵ. Both smooth and discrete geometries are described by integrable systems. It is shown that one can obtain an order ϵ2 approximation globally with points of the discrete nets on the smooth surface. A new simple geometric construction of principal directions of smooth surfaces is given.We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets (circular nets and planar quadrilateral nets, respectively) with edges of order ϵ. Both smooth and discrete geometries are described by integrable systems. It is shown that one can obtain an order ϵ2 approximation globally with points of the discrete nets on the smooth surface. A new simple geometric construction of principal directions of smooth surfaces is given.

New Error Tolerant Method for Search of Long Repeats in DNA Sequences

A new method to identify all sufficiently long repeating nucleotide substrings in one or several DNA sequences is proposed. The method based on a specific gauge applied to DNA sequences that guarantees identification of the repeating substrings. The method allows the matching substrings to contain a given level of errors. The gauge is based on the development of a heavily sparse dictionary of repeats, thus drastically accelerating the search procedure. Some biological applications illustrate the method.

Fast Algorithm for Vernier Search of Long Repeats in DNA Sequences with Bounded Error Density

A novel algorithm to find all sufficiently long repeating nucleotide substrings in one or several DNA sequences is proposed. The algorithm searches approximately matching strings very fast with given level of local error density. Some biological applications illustrate the method.

Universal coefficients for precise interpolation of GNSS orbits from final IGS SP3 data

This paper describes a new framework for very accurate linear interpolation for GLONASS and GPS satellite orbits. Using only 6 consecutive SP3 positions with 15-min time step one may find satellite positions between these points with RMS error 0.4 cm for GPS satellites and 0.2 cm for GLONASS satellites.

Onset of convection in a two–phase binary mixture with the Soret effect in weightlessness

The linear stability of mechanical equilibrium in a two–layer system formed by different phases of the same binary mixture is investigated. The temperature difference is applied to the layers by heating and cooling the opposite rigid boundaries. In the state of mechanical equilibrium, the applied temperature gradient induces concentration gradients due to the Soret effect. The conservation of mass for the mixture components leads to the dependence of layer thicknesses on the applied temperature difference. In weightlessness, the main mechanisms of instability in the considered system are related to phase change and Marangoni effect. The calculations are performed for cyclohexane–methanol binary mixture, which has a liquid–liquid miscibility gap. The analytical solution of amplitude equations for monotonic perturbations is found and expression for the critical temperature difference is derived. It is shown that the phase change instability is long–wave and favoured when the difference between interfacial concentrations in the basic state decreases. When the Marangoni effect is taken into account, additional monotonic and oscillatory modes appear. They result from the interplay between thermocapillarity and phase change with latent heat release/absorption. The most unstable monotonic and oscillatory modes are identified depending on the heating regime and relative thickness of the layers.