On generalisations of Calogero-Moser-Sutherland quantum problem and WDVV equations

It is proved that if the Schr\"odinger equation $L \psi = \lambda \psi$ of Calogero-Moser-Sutherland type with $$L = -\Delta + \sum\limits_{\alpha\in{\cal A}_{+}} \frac{m_{\alpha}(m_{\alpha}+1) (\alpha,\alpha)}{\sin^{2}(\alpha,x)}$$ has a solution of the product form $\psi_0 = \prod_{\alpha \in {\cal {A}_+}} \sin^{-m_{\alpha}}(\alpha,x),$ then the function $F(x) =\sum\limits_{\alpha \in \cal {A}_{+}} m_{\alpha} (\alpha,x)^2 {\rm log} (\alpha,x)^2$ satisfies the generalised WDVV equations.Comment: 10 page

Quantum integrability of the deformed elliptic Calogero-Moser problem

The integrability of the deformed quantum elliptic Calogero-Moser problem introduced by Chalykh, Feigin and Veselov is proven. Explicit recursive formulae for the integrals are found. For integer values of the parameter this implies the algebraic integrability of the systems.Comment: 23 page

N=4 Mechanics, WDVV Equations and Polytopes

N=4 superconformal n-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial nonlinear differential equations generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F. The solutions are encoded by the finite Coxeter systems and certain deformations thereof, which can be encoded by particular polytopes. We provide A_n and B_3 examples in some detail. Turning on the prepotential U in a given F background is very constrained for more than three particles and nonzero central charge. The standard ansatz for U is shown to fail for all finite Coxeter systems. Three-particle models are more flexible and based on the dihedral root systems.Comment: Talk at ISQS-17 in Prague, 19-21 June 2008, and at Group-27 in Yerevan, 13-19 August 2008; v2: B_3 examples correcte

Legal regulation of interreligious relations in the field of general education: the ratio of public and private interests

In article on the basis of the formal legal analysis of the national legal system and the international jurisprudence the key principles of state legal regulation of the confessional relations in the sphere of the general education are distinguished; the need of their addition and unification at the national level for the purpose of providing the balanced ratio of public and private interests in the context of providing the integrated rights and personal freedoms is demonstrate

Spectral Difference Equations Satisfied by KP Soliton Wavefunctions

The Baker-Akhiezer (wave) functions corresponding to soliton solutions of the KP hierarchy are shown to satisfy eigenvalue equations for a commutative ring of translational operators in the spectral parameter. In the rational limit, these translational operators converge to the differential operators in the spectral parameter previously discussed as part of the theory of "bispectrality". Consequently, these translational operators can be seen as demonstrating a form of bispectrality for the non-rational solitons as well.Comment: to appear in "Inverse Problems

Maximal Abelian Subgroups of the Isometry and Conformal Groups of Euclidean and Minkowski Spaces

The maximal Abelian subalgebras of the Euclidean e(p,0) and pseudoeuclidean e(p,1)Lie algebras are classified into conjugacy classes under the action of the corresponding Lie groups E(p,0) and E(p,1), and also under the conformal groups O(p+1,1) and O(p+1,2), respectively. The results are presented in terms of decomposition theorems. For e(p,0) orthogonally indecomposable MASAs exist only for p=1 and p=2. For e(p,1), on the other hand, orthogonally indecomposable MASAs exist for all values of p. The results are used to construct new coordinate systems in which wave equations and Hamilton-Jacobi equations allow the separation of variables.Comment: 31 pages, Latex (+ latexsym

Molecular Mobility in a Poly(ethylene glycol)-Poly(vinyl pyrrolidone) Blends: Study by the Pulsed Gradient NMR Techniques

The molecular mobility in PEG-PVP blends as a function of the time of system storage and the PVP molecular mass is studied by the pulsed-field gradient NMR method. The distribution of PEG molecules over their mobilities is found in a blend containing 36 vol % of PEG with the molecular mass of 400 g/mol. As the storage time of the system increases, the spectrum of diffusion coefficient values varies, thereby indicating the redistribution of PEG400 molecules in the blend with PVP. An anomalous (partly restricted) diffusion of PEG400 molecules is discovered, reflecting the influence of PVP macromolecules on the motion of short PEG chains. It is shown that, during the redistribution of PEG molecules in the blend, they are involved in a complex with PVP, which is characterized by its own transport properties. The data obtained by the NMR relaxation technique are in agreement with the results of NMR diffusion measurements in the studied systems

BPS States in Omega Background and Integrability

We reconsider string and domain wall central charges in N=2 supersymmetric gauge theories in four dimensions in presence of the Omega background in the Nekrasov-Shatashvili (NS) limit. Existence of these charges entails presence of the corresponding topological defects in the theory - vortices and domain walls. In spirit of the 4d/2d duality we discuss the worldsheet low energy effective theory living on the BPS vortex in N=2 Supersymmetric Quantum Chromodynamics (SQCD). We discuss some aspects of the brane realization of the dualities between various quantum integrable models. A chain of such dualities enables us to check the AGT correspondence in the NS limit.Comment: 48 pages, 10 figures, minor changes, references added, typos correcte

Полимеры в лечении ран: реалии и горизонты

Today, biopolymers and biomaterials are a broad and growing sphere of interest to both specialists from various fields of science and to society as a whole. The paper discusses the issues concerning the terminological interpretation of concepts, such as polymers, biodegradability, gels and hydrogels, hydrocolloids, films, sponges, and polymer matrices. It gives the results of the authors’ investigations of bandages as films and porous sponges containing the silver levels as given in the instruction (11 prototypes from different Russian and foreign manufacturers). Electron scanning microscopy revealed a varying content of the ground substance in the prototypes within the range from 0 (even no traces of silver were found) to 7.29 %. Thus, the effect of the test samples of bandages may not be always related to the active component that is positioned the ground components, and the therapeutic effect is achieved, for example, by the polymer base of this dressing agent. The main directions in further investigations of polymers for medical purposes are to elaborate technological approaches to producing biopolymers for the reduction of their cost and to make biomaterials with tailor-made properties. Сегодня биополимеры и биоматериалы – это широкая и активно развивающаяся сфера, интересная как для специалистов из различных областей наук, так и для общества в целом. В статье обсуждаются вопросы терминологической интерпретации таких понятий, как биополимеры, биоразлагаемость, гели и гидрогели, гидроколлоиды, пленки, губки, полимерные матриксы. Представлены результаты собственных исследований перевязочных средств в виде пленок и пористых губок с заявленным в инструкции содержанием серебра (11 опытных образцов от различных отечественных и зарубежных производителей). По данным электронной сканирующей микроскопии выявлено различное содержание основного вещества в опытных образцах в диапазоне от 0 (не обнаружено даже следов серебра) до 7,29 %. Таким образом, действие проанализированных образцов перевязочных средств не всегда может быть связано с тем «активным» компонентом, который позиционируется основным, а лечебный эффект достигается, например, за счет полимерной основы данного перевязочного материала. Основные направления дальнейших исследований полимеров для медицинских целей: разработка технологических подходов производства биополимеров для снижения их стоимости и получение биоматериалов с заданными свойствами.