Quadratic algebras related to elliptic curves

We construct quadratic finite-dimensional Poisson algebras and their quantum versions related to rank N and degree one vector bundles over elliptic curves with n marked points. The algebras are parameterized by the moduli of curves. For N=2 and n=1 they coincide with the Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie-Poisson structure on the direct sum of n copies of sl(N). The derivation is based on the Poisson reduction from the canonical brackets on the affine space over the cotangent bundle to the groups of automorphisms of vector bundles.Comment: 21 page

Modeling of Slow Plasticity Waves

Quasi-static uniaxial loading of a bar with a length L is considered. Mechanical properties of a material in a point are defined by the segment of negative slope on stress-strain diagram which follows the section of elastic deformation. The deformation in specimen is uniform until the stress exceeds the peak yielding stress. The analytical solution shows that stress-strain diagram of the specimen has a yielding plateau. It is shown that the time for a slow wave to advance by a distance equal to the localized band width S is the same as it is required for a plastic wave to run along the whole bar length

Elliptic Schlesinger system and Painlev{\'e} VI

We construct an elliptic generalization of the Schlesinger system (ESS) with positions of marked points on an elliptic curve and its modular parameter as independent variables (the parameters in the moduli space of the complex structure). ESS is a non-autonomous Hamiltonian system with pair-wise commuting Hamiltonians. The system is bihamiltonian with respect to the linear and the quadratic Poisson brackets. The latter are the multi-color generalization of the Sklyanin-Feigin-Odeskii classical algebras. We give the Lax form of the ESS. The Lax matrix defines a connection of a flat bundle of degree one over the elliptic curve with first order poles at the marked points. The ESS is the monodromy independence condition on the complex structure for the linear systems related to the flat bundle. The case of four points for a special initial data is reduced to the Painlev{\'e} VI equation in the form of the Zhukovsky-Volterra gyrostat, proposed in our previous paper.Comment: 16 pages; Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 190

Влияние льдообразования в трещинах на поле температур в холодном слое ледника

The work focuses on modeling the warming of a glacier due to heat release during the refreezing of meltwater in glacier crevasses (cryo-hydrologic warming). The simulation is performed for a polythermal Arctic glacier with a regular network of crevasses filled with water at 0 °C, for the1-year period of freezing of water in crevasses in the cold layer of a glacier, below the active layer. The upper (active layer base) and lower (initial cold-temperate transition surface) boundaries of the cold layer are considered horizontal planes; the crevasses are assumed to be identical narrow straight parallel water-filled channels. These assumptions allow considering the corresponding mathematical problem in a 2D setting. The time-dependent temperature distribution in the modeled domain is calculated explicitly as the solution to a 2D initial boundary value problem for the heat equation with spatially distributed heat sources that model the network of crevasses. The initial temperature distribution and the spatial parameters of the model are set based on the field data from the polythermal glacier Austre Grønfjordbreen (Svalbard). For a fixed geometry of the crevasses (the distance between neighboring crevasses is 10 m, the depth is 10 m, the width is of order 0.1 m) we performed an analytical-solution-based simulation of the temperature field at the end of a year-long period of heating varying the active layer base temperature (-3, -2 °C) and the initial thickness of the cold layer (20, 40, 60 m). The results suggest that the temperature field is more influenced by the cold layer thickness than the upper boundary temperature. The maximum temperature increment is 1–2 °C depending on the simulated case. The cold-temperate transition surface shifts up under the crevasse area by a maximum of 3.4 m (only in the case of 20-m cold layer). The temperature field remains unperturbed at a distance of 20 m or more in any direction from the crevasse zone. Our results may be useful for quantitative comparison of cryo-hydrologic warming with other factors of the temperature state of glaciers.Для условий ледника Восточный Грёнфьорд (Западный Шпицберген) выполнено моделирование изменений температурного поля в холодном слое в конце годового периода отепления за счёт замерзания воды в трещинах. Показано, что температура в леднике повысится на 1–2 °C, граница холодного и тёплого льда сместится вверх максимум на 3,4 м, возмущения температурного поля распространятся не далее 20 м от области трещиноватости