81 research outputs found
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 or 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 of
Calogero-Moser-Sutherland type with
has a solution of the product form then the function 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 superanalog
of Calogero operator is constructed; it is related with the root system of the
Lie superalgebra . For we obtain the usual Calogero
operator; for we obtain, up to a change of indeterminates and parameter
the operator constructed by Veselov, Chalykh and Feigin [2,3]. For the operator is the radial part of the 2nd
order Laplace operator for the symmetric superspaces corresponding to pairs
and , respectively. We will show
that for the generic and the superanalogs of the Jack polynomials
constructed by Kerov, Okunkov and Olshanskii [5] are eigenfunctions of
; for 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
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