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We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra $gl_N$. The models are defined on a tensor product irreducible $gl_N$-modules. For each model, there exist $N$ one-parameter families of commuting operators on the tensor product, called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues., Comment: 48 pages, amstex.tex (ver 2.2), misprints corrected

Under certain conditions, we give an estimate from above on the number of differential equations of order $r+1$ with prescribed regular singular points, prescribed exponents at singular points, and having a quasi-polynomial flag of solutions. The estimate is given in terms of a suitable weight subspace of the tensor power $U(\n_-)^{\otimes (n-1)}$, where $n$ is the number of singular points in $\C$ and $U(\n_-)$ is the enveloping algebra of the nilpotent subalgebra of $\glg_{r+1}$., Comment: Latex, 11 pages, revised version

We prove the B. and M. Shapiro conjecture that says that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This in particular implies the following result: If all ramification points of a parametrized rational curve $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset CP^r $. The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has a real spectrum. In Appendix we discuss properties of differential operators associated with Bethe vectors in the Gaudin model and, in particular, prove a conditional statement: we deduce the transversality of certain Schubert cycles in a Grassmannian from the simplicity of the spectrum of the Gaudin Hamiltonians., Comment: Latex, 18 pages, revised version

Let $V = < p_{ij}(x)e^{\la_ix}, i=1,...,n, j=1, ..., N_i >$ be a space of quasi-polynomials of dimension $N=N_1+...+N_n$. Define the regularized fundamental operator of $V$ as the polynomial differential operator $D = \sum_{i=0}^N A_{N-i}(x)\p^i$ annihilating $V$ and such that its leading coefficient $A_0$ is a polynomial of the minimal possible degree. We construct a space of quasi-polynomials $U = < q_{ab}(u)e^{z_au} >$ whose regularized fundamental operator is the differential operator $\sum_{i=0}^N u^i A_{N-i}(\partial_u)$. The space $U$ is constructed from $V$ by a suitable integral transform. Our integral transform corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) to the KP hierarchy, see \cite{W}. As a corollary of the properties of the integral transform we obtain a correspondence between critical points of the two master functions associated with the $(\glN,\glM)$ dual Gaudin models as well as between the corresponding Bethe vectors.

We generalize the results of [KMST] concerning equivariant quantization by means of Verma modules $M(\lambda)$ for generic weight $\lambda$ to the case of general $\lambda$. We consider the relationship between the Shapovalov form on an irreducible highest weight module of a semisimple complex Lie algebra, fusion elements, and equivariant quantization. We also discuss some limiting properties of fusion elements. [KMST] E. Karolinsky and A. Stolin, Dynamical Yang-Baxter equations, quasi-Poisson homogeneous spaces, and quantization, Lett. Math. Phys., 71 (2005), p.179-197; e-print math.QA/0309203., Comment: Latex, 19 pages

An energy-balance model of global climate, which takes into account a nontrivial role of galactic cosmic rays, is developed. The model is described by the fold catastrophe equation relative to increment of temperature, where galactic cosmic rays and insolation are control parameters. The comparison of the results of a computer simulation of time-dependent solution of the presented model and oxygen isotope records of deep-sea core V28-238 over the past 730 kyr are presented. The climate evolution in future 100 kyr is also predicted., Comment: 11 pages, 3 figures

The reaction pp -> dpi+eta has been measured at a beam energy of T=2.65 GeV (p=3.46 GeV/c) using the ANKE spectrometer at COSY-Juelich. The missing mass distribution of the detected dpi+ pairs exhibits a peak around the eta mass on top of a strong background of multi-pion pp -> dpi+(n(pi)) events. The differential cross section d^4(sigma)/d(Omega_d)d(Omega_pi+)d(p_d)d(p_pi+) for the reaction pp -> dpi+eta has been determined model independently for two regions of phase space. Employing a dynamical model for the a0+ production allows one then to deduce a total cross section of sigma(pp -> da0+ -> dpi+eta)=(1.1 +/- 0.3_(stat) +/- 0.7_(syst)) microbarn for the production of pi+eta via the scalar a0+(980) resonance and sigma(pp -> dpi+eta) = (3.5 +/- 0.3_(stat) +/- 1.0_(syst)) microbarn for the non-resonant production. Using the same model as for the interpretation of recent results from ANKE for the reaction pp -> dK+(bar(K0)), the ratio of the total cross sections is sigma(pp -> d(K+(bar(K0)))_(L=0))/sigma(pp -> da0+ -> dpi+eta) = 0.029 +/- 0.008_(stat) +/- 0.009_(syst), which is in agreement with branching ratios in the literature., Comment: 5 pages, 8 figures, 1 table

Stability with respect to neutron emission is studied for highly neutron-excessive Oxygen isotopes in the framework of Hartree-Fock-Bogoliubov approach with Skyrme forces Sly4 and Ska. Our calculations show increase of stability around 40O., Comment: 5 pages, 3 figures

We consider the reaction g+d -> pi+d in a wide energy range around and above the eta-meson photoproduction threshold at backward CM angles of the outgoing pion. Our theoretical analysis is essentially motivated by the recent measurements of the CLAS Collaboration at Jefferson Lab, where this kinematical region of the reaction has been thoroughly studied for the first time and a cusps in the energy dependence of the differential cross section in the region of Eg = 600-800 MeV has been observed. Our preliminary and qualitative analysis, based on single- and double-scattering diagrams, shows that the observed structure can be explained by the contribution of the double-scattering diagram with intermediate production of the eta meson. The effect, to a considerable extent, is formed due to the contribution of N(1535) resonance to the amplitudes of subprocesses on the nucleons., Comment: 16 pages, 7 figures, submitted to PRC

The energy-balance model of global climate, which is taking into account a nontrivial role of solar and galactic protons, is presented. The model is described by the equation of fold catastrophe relative to increment of temperature, where the variation of a solar insolation and cosmic rays are control parameters. It is shown that the bifurcation equation of the model describes one of two stable states of the climate system. The solution of this equation exhibits the property of the determined bistable behavior of climate at the global level and the possibility of appearance of the determined chaos of "the weathers" at the local levels. The results of the comparative analysis of the computer simulation of the time-dependent solution of energy-balance model of global climate and the oxygen isotope records for deep-sea core V28-238 over the past 730 kyr are presented, and the evolution of climate on 100 kyr forward is also predicted. It is shown that the proposed model successfully explains the nicety of the paleoclimatic records. The model is clear of all known difficulties of the Milankovich theory for the analysis and the interpretation of physical mechanism, by which the climate system responds to orbital forcing., Comment: 21 pages, 6 figures

For the Lie algebra $gl_N$ we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the $gl_N$ rational quantized Knizhnik-Zamolodchikov difference operators. We describe the transformations of the dynamical operators under the natural action of the $gl_N$ Weyl group., Comment: 7 pages, AmsLaTeX

Using the well-known experimental data the inverse problem of neutrino diagnostics of reactor core is considered. The solution of this problem makes it possible to determine distantly the current value of nuclear density of each nuclear fuel components with known accuracy and also opens up the possibility for the development of neutrino online technology of temporal evolution of nuclear fuel isotopic structure and reactor power., Comment: 17 pages, 3 figures

The problem of the geoantineutrino deficit and the experimental results of the interaction of uranium dioxide and carbide with iron-nickel and silica-alumina melts at high pressure (5-10 GPa) and temperature (1600- 22000 C) have induced us to consider the possible consequences of made by V. Anisichkin and A. Ershov supposition that there is an actinoid shell on boundary of liquid and solid phases of the Earth's core. We have shown that the activation of a natural nuclear reactor operating as the solitary waves of nuclear burning in 238U- and/or 232Th-medium (in particular, the neutron-fission progressive wave of Feoktistov and/or Teller-Ishikawa-Wood) such physical consequent can be. The simplified model of the kinetics of accumulation and burnup in U-Pu fuel cycle of Feoktistov is developed. The results of the numerical simulation of neutron-fission wave in two-phase UO2/Fe medium on a surface of the Earth's solid core are presented. On the basis of O'Nions-Ivensen-Hamilton model of the geochemical evolution of mantle differentiation and the Earth's crust growth supplied by actinoid shell on the boundary of liquid and solid phases of the Earth's core as a nuclear energy source, the tentative estimation of intensity and geoantineutrino spectrum on the Earth surface was obtained., Comment: 26 pages, 7 figures

We review results on the Knizhnik-Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the $(gl_k,gl_n)$ duality, and their implications for hypergeometric integrals. The KZ and dynamical equations naturally exchange under the duality, which provides two kinds of integral formulae for hypergeometric solutions of the equations. This fact yields identities for hypergeometric integrals of different dimensions, the dimension of the first integral being the parameter of the second one, and vice versa. Similar identites exist between hypergeometric and q-hypergeometric integrals of Mellin-Barnes type of different dimensions. Those identites are multidimensional analogues of the equality of two integral representations for the Gauss hypergeometric function., Comment: Preprint, 29 pages, AmsLaTeX, to appear in Proceedings of IDAQUIS, (Infinite Dimensional Algebras and Quantum Integrable Systems), Faro, Portugal, July 21-25, 2003

An analytic formula for the two-loop QCD correction of the decay H --> 2 gamma is presented. To evaluate all master integrals a `Risch-like' algorithm was exploited., Comment: 5 pages, Latex, axodraw.sty

The estimations of integral intensity and energy spectrum of geoantineutrino on the Earth's surface (in the absence of oscillations) from different radioactive sources (U, Th and K) are obtained on the basis of temporal evolution analysis of radiogenic heat-evolution power of the Earth w4thin the framework of model of geochemical processes of mantle differentiation and earth's crust growth., Comment: 17 pages, 5 figures

Given complex numbers $m_1,l_1$ and nonnegative integers $m_2,l_2$, such that $m_1+m_2=l_1+l_2$, for any $a,b=0, ... ,\min(m_2,l_2)$ we define an $l_2$-dimensional Barnes type q-hypergeometric integral $I_{a,b}(z,\mu;m_1,m_2,l_1,l_2)$ and an $l_2$-dimensional hypergeometric integral $J_{a,b}(z,\mu;m_1,m_2,l_1,l_2)$. The integrals depend on complex parameters $z$ and $\mu$. We show that $I_{a,b}(z,\mu;m_1,m_2,l_1,l_2)$ equals $J_{a,b}(e^\mu,z;l_1,l_2,m_1,m_2)$ up to an explicit factor, thus establishing an equality of $l_2$-dimensional q-hypergeometric and $m_2$-dimensional hypergeometric integrals. The identity is based on the $(gl_k,gl_n)$ duality for the qKZ and dynamical difference equations., Comment: Preprint (2003), 14 pages, AmsLaTeX, references updated

We give a construction of creation operators responsible for appearance of Bethe vectors with the same eigenvalues of the transfer-matrix for the inhomogeneous arbitrary spin XXZ model at roots of unity with particular quasiperiodic boundary conditions. This construction generalizes the similar one, recently obtained by K.Fabricius and B.M.McCoy, for the homogeneous six-vertex model with periodic boundary conditions. Even in the last case, the given proof for the main formulae are simpler than the original one., Comment: 9 pages, amstex 2.2 and amssym.tex required

It is shown that solutions of the Bethe ansatz equations for the inhomogeneous arbitrary spin XXX or XXZ model satisfy certain identites which generalize those, recently obtained by K.Fabricius and B.M.McCoy, for solutions of the Bethe ansatz equations for the six-vertex model. Even in the last case, the given proof is simpler than the original one., Comment: 4 pages, amstex 2.2 and amssym.tex required

Given complex numbers $m_1,l_1$ and positive integers $m_2,l_2$, such that $m_1+m_2=l_1+l_2$, we define $l_2$-dimensional hypergeometric integrals $I_{a,b}(z;m_1,m_2,l_1,l_2)$, $a,b=0,...,\min(m_2,l_2)$, depending on a complex parameter $z$. We show that $I_{a,b}(z;m_1,m_2,l_1,l_2)=I_{a,b}(z;l_1,l_2,m_1,m_2)$, thus establishing an equality of $l_2$ and $m_2$-dimensional integrals. This identity allows us to study asymptotics of the integrals with respect to their dimension in some examples. The identity is based on the $(gl_k,gl_n)$ duality for the KZ and dynamical differential equations., Comment: Preprint (2003), 9 pages, AmsLaTeX, misprints corrected

The reactions pp\to d\pi^+\eta and pn\to d\pi^0\eta are of special interest for investigating the a_0(980) (J^P=0^+) resonance in the process NN \to da_0 \to d\pi\eta. We study some aspects of those reactions within a general formalism and also in a concrete phenomenological model. In particular, it is shown that the presence of nonresonant (i.e. without excitation of the a_0 resonance) contributions to these reactions yields nonvanishing values for specific polarization observables, i.e. to effects like those generated by a_0-f_0 mixing. An experimental determination of these observables for the reaction pp\to d\pi^+\eta would provide concrete information on the magnitude of those nonresonant contributions to \pi\eta production. We discuss also the possibility of extracting information about a_0-f_0 mixing from the reaction pn \to d\pi^0\eta with polarized proton beam., Comment: 14 pages, 3 figures

We present several formulae for the Selberg type integrals associated with the Lie algebra $sl_3$., Comment: Errata added; 13 pages, amstex.tex 2.2 and amssym.tex required

We discuss rescattering corrections to the impulse approximation for the processes $\pi^- d\to\pi^- d$ and $\gamma d\to\pi^0 d$. It is shown that the rescattering effects (RE) give non-negligible contribution to the real part of these amplitudes. At the same time the contributions from the imaginary parts of impulse and rescattering corrections drastically cancel each other. This cancellation means that the processes $\pi^- d\to \pi^0 nn$ and $\gamma d\to \pi^+ nn / \pi^- pp$ are strongly suppressed near threshold as required by the Pauli principle., Comment: LaTeX, 7 pages, 2 eps figures, 1 table

We study the meromorphic family of intertwining operators between Verma modules and their products with finite-dimensional ones. A regularizing operator, acting in a finite dimensional module U, makes this family holomorphic, and conjugates the dynamical Weyl group operators to constant operators. We prove uniqueness property for regularizing operators, and construct them explicitly for the Lie algebra sl_3., Comment: 35 pages, AMSLaTeX

By the use of the variational method with exponential trial functions the upper and lower bounds of energy are calculated for a number of non-relativistic three-body Coulomb and nuclear systems. The formulas for calculation of upper and lower bounds for exponential basis are given, the lower bounds for great part of systems were calculated for the first time. By comparison of calculations for different bases the efficiency of exponential trial functions and their universality in respect to masses of particles and interaction are demonstrated. The advantage of exponential basis manifests mostly evident for the systems with comparable masses, though its use in one-center and two-center problems is justified too. For effective solution of two-center problem a carcass modification of the trial function is proposed. The stability of various three-particle Coulomb systems is analyzed., Comment: latex, 15 pages, 5 figures, 2 tables (corrected figure which prevents pdf creation)

Some aspects of a0-f0 mixing effects in the reaction ${\vec p} n\to da^0_0$ with perpendicular polarized proton beam are discussed. An angular--asymmetry parameter $A$ is defined to study those effects. It is shown that, for energies close to the production threshold, the angular--asymmetry parameter $A(\theta, \phi)$ is proportional to the a0-f0 mixing amplitude for arbitrary polar and azimuthal angles $\theta$ and $\phi$ of the outgoing $a_0$ meson. This statement is also valid for arbitrary energies, but then only at polar angles $\theta=0^0$ and $\theta=90^0$. The mass dependence of the differential cross section $d\sigma/dm_{\pi^0\eta}$ in the reaction $pn\to d\pi^0\eta$ in the presence of \mix mixing is also discussed., Comment: 17 pages, 3 Figures

We report on the 13C spin-lattice relaxation times in polycristalline fullerite C60 measured over the temperature range 295 K to 1000 K. At temperatures above 470 K, spin-lattice relaxation is dominated by the spin-rotation interaction. From the analysis of temperature dependence of T1(13C), the spin-rotation constant is determined using J- and M-models of rotational diffusion: C(J)=-51 Hz, C(M)=-29 Hz., Comment: 4 pages, 3 figures, submitted to PRL

We consider the Knizhnik-Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the (gl_k,gl_n) duality. We show that the KZ and dynamical equations naturally exchange under the duality., Comment: Final version, 12 pages, amstex.tex 2.2 and amssym.tex required

The differential spectra in the reaction $pp\to K^+\bar K^0 d$ for different reasonable versions of the production amplitude are discussed at the initial proton energy $T_p=2.645$ GeV., Comment: 7 pages, LaTeX, 1 PostScript figure

We discuss the experimental and theoretical status of charge-symmetry violation (CSV) in the elastic scattering of pi+ and pi- on 3H and 3He. Analysis of the experimental data for the ratios r1, r2, and R at Tpi = 142, 180, 220, and 256 MeV provides evidence for the presence of CSV. We describe pion scattering from the three-nucleon system in terms of single- and double-scattering amplitudes. External and internal Coulomb interactions as well as the Delta-mass splitting are taken into account as sources of CSV. Reasonable agreement between our theoretical calculations and the experimental data is obtained for Tpi = 180, 220, and 256 MeV. For these energies, it is found that the Delta-mass splitting and the internal Coulomb interaction are the most important contributions for CSV in the three-nucleon system. The CSV effects are rather sensitive to the choice of pion-nuclear scattering mechanisms, but at the same time, our theoretical predictions are much less sensitive to the choice of the nuclear wave function. It is found, however, that data for r2 and R at Tpi = 142 MeV do not agree with the predictions of our model, which may indicate that there are additional mechanisms for CSV which are important only at lower energies., Comment: 26 pages of RevTeX, 16 postscript figures

The present communication is a continuation of the review, the first part of which was presented in nucl-th/0109033. The second part dealswith the manifistation of Chaotic dynamics at quantum level. The variations of statistical properties of energy spectrum in the process of R-C-R transition have been studied in detail. We proved that the type of the classical motion is correlated with the structure of the eigenfunctions of highly excited states in the R-C-R transition. Shell structure destruction induced by the increase of nonintegrable perturbation was analyzed., Comment: 53 pages, LaTex, 21 figures

We have presented a complete description of classical dynamics generated by the Hamiltonian of quadrupole nuclear oscillations and identified those peculiarities of quantum dynamics that can be interpreted as quantum manifestations of classical stochasticity. Particular attention has been given to investigation of classical dynamics in the potential energy surface with a few local minima. A new technique is suggested for determination of the critical energy of the transition to chaos. It is simpler than criteria of transition to chaos connected with one or another version of overlap resonances criterion. We have numerically demonstrated that for potential with a localized unstable region motion becomes regular at the high energy again, i.e. transition regularity-chaos-regularity {R - C - R} takes place for these potentials. The variations of statistical properties of energy spectrum in the process of {R - C - R) transition have been studied in detail. We proved that the type of the classical motion is correlated with the structure of the eigenfunctions of highly excited states in the (R - C - R) transition. Shell structure destruction induced by the increase of nonintegrable perturbation was analyzed., Comment: 47 pages, LaTex, 17 figures

It is shown that, if isospin is not conserved in $a^0_0$- and $f_0$-meson mixing, forward-backward asymmetry arises in the reaction $pn\to da^0_0$. This effect increases near the reaction threshold. The asymmetry is estimated within the framework of a model in which the $a^0_0$-$f_0$ mixing is due to the virtual $\pi^0\leftrightarrow\eta$ transition and the difference in masses of the charged and neutral kaons in decay channels. The angular asymmetry near the threshold of the $pn\to da^0_0$ reaction was found to be large, of the order of $8\div 15$ %., Comment: 8 pages, LaTeX, 4 PostScript figures

The small elliptic quantum group $e_{\tau,\gamma}(sl_N)$, introduced in the paper, is an elliptic dynamical analogue of the universal enveloping algebra $U(sl_n)$. We define highest weight modules, Verma modules and contragradient modules over $e_{\tau,\gamma}(sl_N)$, the dynamical Shapovalov form for $e_{\tau,\gamma}(sl_N)$ and the contravariant form for highest weight $e_{\tau,\gamma}(sl_N)$-modules. We show that any finite-dimensional $sl_N$-module and any Verma module over $sl_N$ can be lifted to the corresponding $e_{\tau,\gamma}(sl_N)$-module on the same vector space. For the elliptic quantum group $E_{\tau,\gamma}(sl_N)$ we construct the evaluation morphism $E_{\tau,\gamma}(sl_N)\to e_{\tau,\gamma}(sl_N)$, thus making any $e_{\tau,\gamma}(sl_N)$-module into an evaluation $E_{\tau,\gamma}(sl_N)$-module., Comment: 34 pages, amstex.tex (ver. 2.1), misprints are corrected

The trigonometric KZ equations associated with a Lie algebra $\g$ depend on a parameter $\lambda\in\h$ where $\h\subset\g$ is the Cartan subalgebra. We suggest a system of dynamical difference equations with respect to $\lambda$ compatible with the KZ equations. The dynamical equations are constructed in terms of intertwining operators of $\g$-modules., Comment: 23 pages, AmsTeX, third version, some misprints were corrected

We define a system of "dynamical" differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra $\mathbf{g}$. These are equations on a function of $n$ complex variables $z_i$ taking values in the tensor product of $n$ finite dimensional $\mathbf{g}$-modules. The KZ equations depend on the "dual" variable in the Cartan subalgebra of $\mathbf{g}$. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions., Comment: 34 pages, LaTeX