Another realization of the category of modules over the small quantum group

Let $g$ be a semi-simple simply-connected Lie algebra and let $U_\ell$ be the corresponding quantum group with divided powers, where $\ell$ is an even order root of unity. Let in addition $u_\ell\subset U_\ell$ be the corresponding "small" quantum group. In this paper we establish the following relation between the categories of representations of $U_\ell$ and $u_\ell$: We show that the category of $u_\ell$-modules is naturally equivalent to the category of $U_\ell$-modules, which have a {\it Hecke eigen-property} with respect to representations lifted by means of the quantum Frobenius map U_\ell\ti U(\check g), where $g$ is the Langlands dual Lie algebra. This description allows to express the regular linkage class in the category $u_\ell$-mod in terms of perverse sheaves on the affine flag variety with a Hecke eigen-property. Moreover, it can serve as a basis to the program to understand the connection between the category $u_\ell$-mod and the category of representations of the corresponding affine algebra at the critical level.Comment: Corrected versio

Modules over the small quantum group and semi-infinite flag manifold

We develop a theory of perverse sheaves on the semi-infinite flag manifold $G((t))/N((t))\cdot T[[t]]$, and show that the subcategory of Iwahori-monodromy perverse sheaves is equivalent to the regular block of the category of representations of the small quantum group at an even root of unity

Super Tonks-Girardeau state in an attractive one-dimensional dipolar gas

The ground state of a one-dimensional (1D) quantum gas of dipoles oriented perpendicular to the longitudinal axis, with a strong 1/x^3 repulsive potential, is studied at low 1D densities $n$. Near contact the dependence of the many-body wave function on the separation x_{jl} of two particles reduces to a two-body wave function \Psi_{rel}(x_{jl}). Immediately after a sudden rotation of the dipoles so that they are parallel to the longitudinal axis, this wave function will still be that of the repulsive potential, but since the potential is now that of the attractive potential, it will not be stationary. It is shown that as nd^2 -> 0 the rate of change of this wave function approaches zero. It follows that for small values of nd^2, this state is metastable and is an analog of the super Tonks-Girardeau state of bosons with a strong zero-range attraction. The dipolar system is equivalent to a spinor Fermi gas with spin $z$ components \sigma_{\uparrow}=\perp (perpendicular to the longitudinal axis) and \sigma_{\downarrow}=|| (parallel to the longitudinal axis). A Fermi-Fermi mapping from spinor to spinless Fermi gas followed by the standard 1960 Fermi-Bose mapping reduces the Fermi system to a Bose gas. Potential experiments realizing the sudden spin rotation with ultracold dipolar gases are discussed, and a few salient properties of these states are accurately evaluated by a Monte Carlo method.Comment: 5 pages, 2 figures, revtex4. Published versio

Modification the Harris-Wilson's model of managment by Reserves

В статті розглянуто проблемні питання щодо застосування на практиці моделі оптимального розміру замовлення Харріса-Уілсона та запропоновані шляхи подолання обмежень, пов’язаних з умовами її використання. В статье рассмотрены проблемные вопросы применения на практике модели оптимального размера заказа Харриса-Уилсона и предложены пути преодоления ограничений, связанных с условиями её использования. The article deals with issues concerning the application in practice model of optimalorder size Harris-Wilson and proposed ways of overcoming the limitationsassociated with the conditions of its use

Statistical methods in calculating safety stocks

В статье рассмотрены проблемные вопросы, связанные с формированием политики управления запасами предприятия и применения на практике статистических методов при расчете страховых запасов в условиях нестабильности спроса и действий поставщиков. Проанализированы неточности в толковании отдельных терминов и определений и предложены пути их унификации и корректного использования. В статті розглянуто проблемні питання, пов'язані з формуванням політики управління запасами підприємства та застосуванням на практиці статистичних методів при розрахунку страхових запасів в умовах нестабильності попиту та дій постачальника. Проаналізовані неточності в розумінні окремих термінів і визначень та запроповано шляхи щодо їх уніфікайії та коректного використання. The article deals with the problematic issues related to policy-making inventory management of the enterprise and practical application of statistical methods for the calculation of safety stocks in conditions of instability in demand and the actions of suppliers. The errors in the interpretation of certain terms and definitions are analyzed and ways of their harmonization and proper use are offered

Gravitational sedimentation of cloud of solid spherical particles at small Reynolds numbers

The experimental results of study of gravitational sedimentation of highly-concentrated systems of solid spherical particles at small Reynolds numbers Re<1 are presented. Empirical equation for drag coefficient of the particle assembly has been obtained. The influence of initial particle concentration in the cloud on its dynamics and velocity has been analysed