Novel solvable many-body problems

Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type (“acceleration equals forces”) which determine the motion of points in the complex plane. These models are solvable, namely their con?guration at any time can be obtained from the initial data by algebraic operations, amounting to the determination of the zeros of a known time-dependent polynomial in the independent variable z. Some of these models are multiply periodic, isochronous or asymptotically isochronous; others display scattering phenomena.     

On the existence of solutions and averaging of multipoint boundary-value problems for multifrequency systems with linearly transformed argument

For an m-frequency system of differential equations with linearly transformed argument, we prove the existence of a solution that satisfies initial or multipoint boundary conditions. We obtain an estimate for the error of the method of averaging over fast variables that explicitly depends on a small parameter. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 462–471, October–December, 2008.     

Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.     

Modification of the lie‐algebraic scheme and approximation error estimations

A new modification of the the Lie‐algebraic scheme for solving partial differential equations with initial and boundary conditions based on constructing quasirepresentations of the Heisenberg‐Weyl algebra operators involving boundary conditions is proposed. Approximation errors for the modified scheme are evaluated. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generations of Monic Polynomials such that the Coefficients of Each Polynomial of the Next Generation Coincide with the Zeros of a Polynomial of the Current Generation,and New Solvable Many-Body Problems

The notion of generations of monic polynomials such that the coefficients of each polynomial of the next generation coincide with the zeros of a polynomial of the current generation is introduced, and its relevance to the identification of endless sequences of new solvable many-body problems “of goldfish type” is demonstrated.     

A new solvable many-body problem of goldfish type

A new solvable many-body problem of gold?sh type is introduced and the behavior of its solutions is tersely discussed.     

Properties of the Zeros of the Polynomials Belonging to the Askey Scheme

In this paper, we provide properties—which are, to the best of our knowledge, new—of the zeros of the polynomials belonging to the Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted.     

Effect of combined radiation processing on parameters of Si-based MOS transistors

The variation of parameters of Si-based MOS-transistors affected by combined infrared and X- ray photons processing were presented. It was found that X-ray irradiated transistor under incoherent infrared radiation is characterized by lowering of radiation sensitivity. As positive charge and its radiation-induced rate of generation in undergate SiO₂ insulator decreases, we observe a reduction of both ΔU_No and voltage threshold U_th transistor components. Under pulsed laser processing (λ?=?1.06?µm, τ?=?10^?3s) of the X-rays irradiated transistor the energy density of laser beam corresponding to the minimum threshold voltage was established. It was shown that the combined treatment can be effectively used to control the transistor parameters. The model describing the radiation changes of parameters in Si-SiO₂ structure was proposed. The effect of Na⁺, K⁺ and H⁺ ions in undergate insulator on variation of charge state and interface reconstruction under laser irradiation were predicted by proposed model.