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Information × Registration Number 0211U000859, 0106U002561 , R & D reports Title Large dimmentions and macroscopic parameter number effects in mathematical physics and spectral theory popup.stage_title Head Pastur Leonid, Registration Date 28-02-2011 Organization B.Verkin Institute for Low Temperature Physics and Engineering of National Academy of Science of Ukraine popup.description2 The project deals with investigations of the objects describing by infinitely many parameters (variables), which appear in different branches of mathematical physics, in particular, in the theory of disordered systems, infinitedimensional dynamical systems, in partial derivatives equations theory, spectral theory of random and almost-periodical operators and matrices. These problems are in the center of attention of many investigator groups in the world and have being interesting themselves and with their many applications in solid-state physics, combinatorics and information theory. There are the following results obtained: The limiting normalized counting measure of eigenvalues of additive deformed unitary (orthogonal) invariant ensembles is obtained under optimal conditions, and for Gaussian unitary ensembles with additional symmetry. The covariance of Stieltjes transforms of normalized counting measures of deformed unitary (orthogonal) invariant random matrix ensembles is studied. The central limit theorems for linear eigenvalue statistics of real symmetric and hermitian Wigner random matrices and empirical covariance matrices are proved for sufficiently smooth test functions. The explicit form of variance of respecting Gaussian variables is found. The central limit theorems for matrix elements of functions of the Gaussian unitary and orthogonal random matrix ensembles are proved for differentiable test functions with bounded derivative. The law of large numbers is proved and the conditions for central limit theorems validness for U-statistics and von-Mises statistics of eigenvalues of matrices of real symmetric and hermitian Wigner ensembles, empirical covariance matrices and matrix models are found. The behaviour of Jacobi matrices and linear eigenvalue statistics of hermitian matrix models is investigated. It is proved that Jacoby matrices have asymptotically quasi-periodical coefficients, which defined by the geometry of equilibrium measure support. The central limit theorems for linear eigenvalue statistics of real symmetric matrix models and adjacency matrices of large dimensional random graphs are proved. The local eigenvalue distribution inside the spectrum (bulk) and near the boundary is investigated for real symmetric and symlectic matrix models under condition that the support of respective equilibrium measure is one interval. The universality of these distributions is proved. The asymptotic decomposition of the logarithm of integral of the eigenvalue distribution for real symmetric and symplectic matrix models is found under condition that the support of respective equilibrium measure is one interval. The local eigenvalue distribution inside the spectrum (bulk) for unitary matrix models is investigated. The universality of this distribution is proved. The asymptotic decomposition of Verblonsky coefficients of orthogonal polynomials on the unit circle with the weight is found under condition that the support of respective equilibrium measure is one arc. The universality of the local eigenvalue distribution inside the spectrum (bulk) for deformed Gaussian ensemble and the covariance matrix ensemble is proved. The dynamical behaviour of the neuron system of large dimension with random non symmetrical interaction is studied. The stability condition of this system is studied. The scattering theory for Jacoby operator on the axis with asymptotical finite-zone coefficients scale-type is founded. The necessary and sufficient condition on the scattering data allowing to solve the respecting direct and reversal problems in the class of perturbations having finite second moments is established. For the Jacoby operators with asymptotically constant coefficients these problems are solved in the class with finite first moments. The associated Cauchy problem for Toda hierarchy is solved. With help of double commutational methods the soliton solutions of these equations are obtained on quasi-periodical backgrounds. The Ramanian spectra of GSU-phases of solid He, Ar, Kr and Xe are calculated and experimentally studied on the wide interval of pressures up to megabares and the detail correspondence with the theory is completed. The dependence of distortion parameter of structure describing deviation of axial relation from ideal value from the pressure for GSU of solid gases is calculated. For t-J Model on rectangular and triangular structures close half-occupation it is shown that the temperature dependence of heat capacity have double peak form. The model describing the thermodynamical properties of many-lattice atomic deposit precipitate on the surface and in the canals of carbon nanotubes is proposed. For Heisenberg ferromagnetic with spin S=1/2 in external magnetic field the approach which describe thermodynamical functions in wide region of temperatures and magnetic fields is proposed. As result of investigations: 51 articles is published; 3 articles is in preparation; 41 conference reports is made; 1 doctoral dissertation is prepared. Product Description popup.authors Єгорова Ірина Євгенівна Анцигіна Тетяна Миколаївна Васильчук Володимир Юрійович Венгеровський Валентин Валентинович Звягін Андрій Анатолійович Литова Ганна Юріївна Пастур Леонід Андрійович Фрейман Юрій Олександрович Щербина Марія Володимирівна popup.nrat_date 2020-04-02 Close