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Panyushev D.I.,Independent University of Moscow
Selecta Mathematica, New Series | Year: 2010

Let G be a simple algebraic group and B a Borel subgroup. We consider generalisations of Lusztig's q-analogues of weight multiplicity, where the set of positive roots is replaced with the multiset of weights of a B-submodule N of an arbitrary finite-dimensional G-module V. The corresponding polynomials in q are called generalised Kostka-Foulkes polynomials (gKF). We prove vanishing theorems for the cohomology of line bundles on G × BN and derive from this a sufficient condition for the non-negativity of the coefficients of gKF. We also consider in detail the case in which V is the simple G-module whose highest weight is the short dominant root and N is the B-submodule whose weights are all short positive roots. © 2010 Birkhäuser / Springer Basel AG.

Mironov D.,Independent University of Moscow | Sossinsky A.,Independent University of Moscow
Russian Journal of Mathematical Physics | Year: 2015

A modified version of the ASEP model is interpreted as a two-dimensional model of ideal gas. Its properties are studied by simulating its behavior in different situations, using an animation program designed for that purpose. © 2015, Pleiades Publishing, Ltd.

Krasil'shchik J.,Independent University of Moscow | Verbovetsky A.,Independent University of Moscow
Journal of Geometry and Physics | Year: 2011

An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with constraints (on a PDE) are discussed. Analogs of tangent and cotangent bundles to a differential equation are introduced and the variational Schouten bracket is defined. General theoretical constructions are illustrated by a series of examples. © 2010 Elsevier B.V.

Mejia-Monasterio C.,University of Helsinki | Mejia-Monasterio C.,Technical University of Madrid | Oshanin G.,University Pierre and Marie Curie | Oshanin G.,Independent University of Moscow
Soft Matter | Year: 2011

A particle driven by an external force in a molecular crowding environment - a quiescent bath of other particles, makes their spatial distribution inhomogeneous: the bath particles accumulate in front of the biased particle (BP) and are depleted behind. In fact, a BP travels together with the inhomogeneity it creates. A natural question is what will happen with two BPs when they appear sufficiently close to each other such that the inhomogeneities around each of them start to interfere? In quest for the answer we examine here, via Monte Carlo simulations, the dynamics of two BPs in a lattice gas of bath particles. We observe that for a sufficiently dense medium, surprisingly, both BPs spend most of the time together which signifies that the interference of the microstructural inhomogeneities results in effectively attractive interactions between them. Such statistical pairing of BPs minimizes the size of the inhomogeneity and hence reduces the frictional drag force exerted on the BPs by the medium. As a result, in some configurations the center-of-mass of a pair of BPs propagates faster than a single isolated BP. These jamming-induced forces are very different from fundamental physical interactions, exist only in presence of an external force, and require the presence of a quiescent bath to mediate the interactions between the driven particles. © 2011 The Royal Society of Chemistry.

Belavin A.A.,Landau Institute for Theoretical Physics | Bershtein M.A.,Independent University of Moscow | Feigin B.L.,Independent University of Moscow | Litvinov A.V.,University of California at Santa Barbara | Tarnopolsky G.M.,University of California at Santa Barbara
Communications in Mathematical Physics | Year: 2013

The recently proposed relation between conformal field theories in two dimensions and supersymmetric gauge theories in four dimensions predicts the existence of the distinguished basis in the space of local fields in CFT. This basis has a number of remarkable properties: one of them is the complete factorization of the coefficients of the operator product expansion. We consider a particular case of the U(r) gauge theory on ℂ2/ℤp which corresponds to a certain coset conformal field theory and describe the properties of this basis. We argue that in the case p = 2, r = 2 there exist different bases. We give an explicit construction of one of them. For another basis we propose the formula for matrix elements. © Springer-Verlag Berlin Heidelberg 2012.

Olshanski G.,Independent University of Moscow
Electronic Journal of Combinatorics | Year: 2010

Let Mn stand for the Plancherel measure on Yn, the set of Young diagrams with n boxes. A recent result of R. P. Stanley (arXiv:0807.0383) says that for certain functions G defined on the set Y of all Young diagrams, the average of G with respect to Mn depends on n polynomially. We propose two other proofs of this result together with a generalization to the Jack deformation of the Plancherel measure.

Bershtein M.A.,Independent University of Moscow | Fateev V.A.,Montpellier University | Litvinov A.V.,Rutgers University
Nuclear Physics B | Year: 2011

In this paper we consider parafermionic Liouville field theory. We study integral representations of three-point correlation functions and develop a method allowing us to compute them exactly. In particular, we evaluate the generalization of Selberg integral obtained by insertion of parafermionic polynomial. Our result is justified by different approach based on dual representation of parafermionic Liouville field theory described by three-exponential model. © 2011 Elsevier B.V.

Avvakumov S.,Independent University of Moscow | Karpenkov O.,University of Liverpool | Sossinsky A.,Independent University of Moscow
Russian Journal of Mathematical Physics | Year: 2013

In this paper, we study normal forms of plane curves and knots. We investigate the Euler functional E (the integral of the square of the curvature along the given curve) for closed plane curves, and introduce a closely related functional A, defined for polygonal curves in the plane ℝ2 and its modified version A R, defined for polygonal knots in Euclidean space ℝ3. For closed plane curves, we find the critical points of E and, among them, distinguish the minima of E, which give us the normal forms of plane curves. The minimization of the functional A for plane curves, implemented in a computer animation, gives a very visual approximation of the process of gradient descent along the Euler functional E and, thereby, illustrates the homotopy in the proof of the classical Whitney-Graustein theorem. In ℝ3, the minimization of A R (implemented in a 3D animation) shows how classical knots (or more precisely thin knotted solid tori, which model resilient closed wire curves in space) are isotoped to normal forms. © 2013 Pleiades Publishing, Ltd.

Sossinsky A.B.,Independent University of Moscow
Russian Journal of Mathematical Physics | Year: 2016

We describe a discrete 3D model of ideal gas based on the idea that, on the microscopic level, the particles move randomly (as in ASEP models), instead of obeying Newton’s laws as prescribed by Boltzmann. © 2016, Pleiades Publishing, Ltd.

Sossinsky A.B.,Independent University of Moscow
Russian Journal of Mathematical Physics | Year: 2012

This article is a continuation of the study of new types of knot energy undertaken in [1, 2] (but is formally independent of those articles); it describes some experiments with mechanical models of knots (that we call twisted wire knots), contains rigorous definitions of their mathematical counterparts, formulations of a series of problems and conjectures. Different energy functionals for various classes of knot types and the corresponding normal forms are discussed and compared. © 2012 Pleiades Publishing, Ltd.

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