On algebraic integrability of the deformed elliptic Calogero--Moser problem

Algebraic integrability of the elliptic Calogero--Moser quantum problem related to the deformed root systems \pbf{A_{2}(2)} is proved. Explicit formulae for integrals are found

A remark on rational isochronous potentials

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form $U(x) = 1/2 \omega^2 x^2$ or $U(x) = 1/8 \omega^2 x ^2 + c^2 x^{-2}.$Comment: 5 pages, contribution to a special issue of JNMP dedicated to F. Calogero, slightly revised versio

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

Multidimensional Baker-Akhiezer functions and Huygens' Principle

A notion of rational Baker-Akhiezer (BA) function related to a configuration of hyperplanes in C^n is introduced. It is proved that BA function exists only for very special configurations (locus configurations), which satisfy certain overdetermined algebraic system. The BA functions satisfy some algebraically integrable Schrodinger equations, so any locus configuration determines such an equation. Some results towards the classification of all locus configurations are presented. This theory is applied to the famous Hadamard's problem of description of all hyperbolic equations satisfying Huygens' Principle. We show that in a certain class all such equations are related to locus configurations and the corresponding fundamental solutions can be constructed explicitly from the BA functions.Comment: 35 pages, LATEX, 2 figures included in graphicx. Submitted to Comm.Math.Phys. (Dec. 1998

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

Superanalogs of the Calogero operators and Jack polynomials

A depending on a complex parameter $k$ superanalog ${\mathcal S}{\mathcal L}$ of Calogero operator is constructed; it is related with the root system of the Lie superalgebra ${\mathfrak{gl}}(n|m)$. For $m=0$ we obtain the usual Calogero operator; for $m=1$ we obtain, up to a change of indeterminates and parameter $k$ the operator constructed by Veselov, Chalykh and Feigin [2,3]. For $k=1, \frac12$ the operator ${\mathcal S}{\mathcal L}$ is the radial part of the 2nd order Laplace operator for the symmetric superspaces corresponding to pairs $(GL(V)\times GL(V), GL(V))$ and $(GL(V), OSp(V))$, respectively. We will show that for the generic $m$ and $n$ the superanalogs of the Jack polynomials constructed by Kerov, Okunkov and Olshanskii [5] are eigenfunctions of ${\mathcal S}{\mathcal L}$; for $k=1, \frac12$ they coinside with the spherical functions corresponding to the above mentioned symmetric superspaces. We also study the inner product induced by Berezin's integral on these superspaces

Duality for Jacobi group orbit spaces and elliptic solutions of the WDVV equations

From any given Frobenius manifold one may construct a so-called dual structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the WDVV equations of associativity. Jacobi group orbit spaces naturally carry the structures of a Frobenius manifold and hence there exists a dual prepotential. In this paper this dual prepotential is constructed and expressed in terms of the elliptic polylogarithm function of Beilinson and Levin

Autodiffusion and phase separation in aqueous solutions of polyoxypropylene diol

Phase equilibrium in the polyoxypropylene-water system at the lower consolute temperature has been studied by an NMR impulse method and by an interference micro-method. An analysis of the shape of the diffusional attenuations which in this case are complex, and also of the experimentally obtained population values, enabled one to construct the phase diagram in the 250-315° K range. The concentrated polymer phase formed by phase separation consists of polyoxypropylene diol, almost free of water. © 1989

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