## Development of the Institute of Mathematics

at the modern stage (1990-2003)

In early 90's, the Institute of Mathematics consisted of 18 scientific departments comprising 10 structural laboratories: the Department of Mathematical Physics and the theory of nonlinear oscillations (supervisor Yu. O. Metropolsky) with laboratories of boundary-value problems of electrodynamics and elasticity (A. A. Berezovsky), information technologies and computer mathematics (V. A. Shirokov), the theory of dynamic systems (O. M. Sharkovsky), ordinary differential equations (A. M. Samoilenko), algebras (A. V. Roiter), topological methods of analysis (Yu. T. Trokhimchuk), Probability Theory and mathematical statistics (V. S. Korolyuk) from the Laboratory of Applied Statistics (A. F. Turbin), functional analysis (Yu. M. Berezansky) from the laboratory of inverse problems of spectral analysis (L. P. Nyzhnik), random processes (A. V. Skorokhod) with a laboratory of stochastic differential equations and diffusion processes (M. I. Portenko), differential equations with partial derivatives (M. L. Gorbachuk), the theory of functions (V. K. Dzyadyk) from the laboratory of harmonic analysis (O. I. Stepanets), theory of approximations (M. P. Korneichuk), complex analysis and potential theory (P. M. Tamrazov), applied researches (V. I. Fushchich) with laboratory of mathematical problems of heat and mass transfer (A. S. Galitsin), mathematical modeling (B. B. Nesterenko), stability of multidimensional systems (I. O. Lukovsky) with a laboratory of mathematical problems of mechanics (D. G. Korenivsky), mechanics and control processes (V. M. Koshlyakov), theory of reliability of probabilistic systems (G. P. Butsan) from the laboratory of statistical methods of the theory of reliability (I. I. Ezhov), mathematical methods of statistical mechanics (D. Ya. Petrina).

During this period, scientists of the Institute had been carrying out research on such important areas of mathematics as algebra, topology, theory of functions, functional analysis, theory of ordinary differential equations and partial differential equations, mathematical physics and theory of nonlinear oscillations, probability theory and mathematical statistics, mathematical methods of mechanics, computational mathematics, mathematical modeling and applied mathematics.

In theory of nonlinear oscillations asymptotic methods were developed for higher order equations and partial derivatives, adiabatic invariants were constructed for broad classes of dynamical systems, and important theorems of the theory of stability were proved (Yu. O. Mitropolsky); significant results were obtained in constructing a constructive theory of local central manifolds (O. B. Lykova).

In the theory of differential equations the problem of asymptotic splitting of a singularly perturbed system of linear differential equations in a complex bifurcation point of system coefficients was solved; have been developed methods of asymptotic integration of linear systems with slowly varying coefficients and degenerations, has been completed an analysis of the numerical-analytic method for the investigation of periodic solutions of nonlinear differential equations, in particular, was found the exact value of the radius of convergence of the majorant series of this method; was constructed a Fawar theory for linear pulse systems with bounded operator coefficients in a Banach space (A. M. Samoilenko).

In 1996, Yu. O. Mitropolsky, A. M. Samoilenko, V. L. Kulik, O. K. Lopatin and M. I. Ronto were awarded the State Prize of Ukraine for the cycle of works "New mathematical methods in nonlinear analysis".

Was developed the basic provisions of the theory of almost periodic pulsed systems and the theory of linear pulsed extensions of dynamical systems on a torus (S. I. Trfymchuk, V. I. Tkachenko).

Was constructed the basis of the local theory of nonlinear functional equations, was developed the method of Norcon forms of PoincarÃ© for nonautonomous difference equations (G. P. Pelyukh).

Were obtained significant results in the theory of noether boundary value problems for systems of differential equations and equations with impulse action (O. A. Boichuk).

In the theory of dynamical systems was proposed the classification of one-dimensional dynamical systems by the type of rotating trajectory; were found the criteria of simplicity and complexity; has been developed a new approach to mathematical modeling of turbulence (the concept of "ideal turbulence") and from the new point of view was considered the development of the cascade process of formation of structures and the emergence of spatial-temporal deterministic chaos; Mathematical formalism was proposed for describing the processes of formation of structures, including fractal, solutions of difference equations with a continuous argument (O. M. Sharkovsky).

In the theory of differential equations with partial derivatives was created a theory of the degree of perturbation of a densely set maximally monotonous operator with its application to the solvability problems of variational inequalities and differential inclusions of elliptic and parabolic types, was constructed a correction for relatively uniform convergence for solving a nonlinear parabolic problem in a general perforated region and studied the behavior of the remainder of its asymptotic expansion (I. V. Skrypnik).

Were studied the evolution equations with a regularized fractional derivative with respect to time, which are widely used in physics to describe abnormal diffusion; was constructed Green's matrix of the Cauchy problem for a nonhomogeneous equation of fractal diffusion with variable coefficients (S. D. Eidelman, A. N. Kochubei).

Was constructed the theory of pseudodifferential operators over the field of p-adic numbers and general local fields; was obtained the image of canonical commutation relations by operators over a local field of characteristic n, which made it possible to systematically develop the fundamentals of analysis and the theory of ordinary differential equations over such fields; was developed the theory of differential equations with irregular singularities over the field of a positive characteristic (A. N. Kochubei).

For differential equations in the Banach space both over the archimedean and non-archi-field fields, were found the criteria of solvability of the Cauchy problem in various classes of analytic vector-functions of finite order and type, by means of which the boundaries of applying the method of power series to finding both exact and approximate solutions of these equations; for approximate solutions are obtained a priori estimations of the approximation error; was constructed the theory of boundary values of semigroups of linear operators in a Banach space, were found criteria for the solvability of differential equations in a Banach space in classes of entire vector-valued functions of finite order (M. L. Gorbachuk, V. I. Gorbachuk).

In the evolution described by differential equations of the second order hyperbolic type was extended the Lax-Phillips scheme; the image of the scattering matrix was obtained and its dependence on the choice of free evolution was investigated; solved the inverse problem (S.O. Kuzhel).

Was developed Non-Gaussian, in particular Poisson's, infinite-dimensional analysis on the spaces conjugate to the nuclear and on the confinement spaces and the spectral theory of the Jacobian fields, on the basis of which was constructed the generalization of the chaotic image for the gamma field of operators and the corresponding stochastic process; the theory of schedules was constructed by compatible generalized own vectors of the general families of commuting normal operators; has been developed the theory of scattering in terms of bilinear functionalisms (Yu. M. Berezansky, L. P. Nyzhnik, V. D. Koshmanenko).

Have been developed the methods of the theory of hypergroups and algebraic methods of functional analysis, which, in particular, made it possible to describe the image of a wide class of quantum groups and homogeneous spaces (Yu. M. Berezansky, Yu. S. Samoilenko).

Was carried out a harmonic analysis on the configurations of spaces and was proposed its application to equilibrium and non-equilibrium problems of infinitely-particle systems of mathematical physics. A modification of biorthogonal analysis was suggested for use in the Poisson analysis (Yu. M. Berezansky, Yu. G. Kondratiev).

Were developed new mathematical methods for investigating equilibrium states in classical and quantum continuous systems and proved their applications to mathematical physics models, the existence of Glauber dynamics for the general class of interaction potentials, and was obtained a new system of equations for the correlation functions of such dynamics; were proposed new approaches to the construction of diffusion and Glauber processes, had been analyzed ergodic properties and scaling boundaries of the processes under consideration (Yu. G. Kondratiev, O. L. Rebenko).

New theoretical and operative methods for the analysis of SchrÃ¶dinger operators with singular potentials have been developed and their spectral properties have been investigated (V. A. Mikhailets).

Were investigated singularly perturbed self-directed operators on the basis of perturbation of bilinear forms, were obtained conditions for the appearance of eigenvalues in the spectral lacunae of the main operator, and were studied the spectral properties of the SchrÃ¶dinger operator with singular potential (L. P. Nyzhnik, V. D. Koshmanenko).

In 1998, for the cycle of works "New methods in the theory of generalized functions and their application to mathematical physics" Yu. M. Berezansky, V. I. Gorbachuk, M. L. Gorbachuk, Yu. G. Kondratiev and L. P. Nyzhnik were awarded State Prize of Ukraine.

In mathematical physics and statistical mechanics were constructed solutions of the Bogolyubov equations for classical and quantum infinite systems, were investigated the spectra of model Hamiltonians in the spaces of translation-invariant functions, Boltzmann's equation were derived from the Bogolyubov equations without the use of additional physical hypotheses, the equations for coefficients of the matrix were found and investigated, were studied scattering of polynomial models, the spectrum of the model Hamiltonian of the theory of superconductivity for a finite cube periodic boundary conditions (D. Ya. Petrina, V. I. Gerasimenko).

Was investigated the hierarchy of the Bogolyubov type diffusion equations describing the Brownian dynamics of planar rotators, oscillators and particle interaction particles; were found generalized solutions of the Hibbs type of these hierarchies, and was solved the Cauchy problem in Banach spaces containing equilibrium (Hibbs) correlation functions (V. I. Skrypnik).

Were developed the methods for constructing solutions for the dual Bogolyubov equation hierarchy for infinite quantum and classical systems of particles (V. I. Gerasimenko).

Was constructed an irreducible image of the paraceralgebra of Poincare, which contains central charges and algebra of internal symmetries; For the first time were found the equations of motion of a relativistic particle with spin 3/2, which do not have non-causal relations; was constructed a parasuperysymmetric model of Weigher - Zumino and a supersymmetric quantum mechanics model with central charges (A. G. Nikitin).

In 2001, for the cycle of works "Functional analytical and group methods in mathematical physics" D. Ya. Petrina, V. I. Gerasimenko, A. G. Nikitin, P. V. Malyshev, V. I. Fushchich (posthumously) were awarded State Prize of Ukraine.

Markov disturbances of differential, integral, and difference equations were considered in probability theory; was found the asymptotic behavior of solutions for high frequency perturbations; was proved a theorem for the existence of a solution of an infinite system of stochastic differential equations that describes the behavior of an infinite number of interacting particles (A. V. Skorokhod).

Were developed asymptotic methods for the analysis of stochastic differential equations, the theory based on the concept of an extended stochastic integral and methods for constructing and studying mathematical models of diffusion phenomena in media with translucent membranes (A. V. Skorokhod, M. I. Portenko).

Were substantiated the heuristic principles of phase consolidation of complex systems, significant results were obtained in the theory of mass service and reliability theory, a number of boundary theorems for semi-Markov processes were proved, and was constructed the Poisson approximation of stochastic homogeneous additive functionals with semimarkoff switches (V. S. Korolyuk).

On the basis of new mathematical models of the diffusion phenomenon in translucent membranes was studied the behavior of particles diffusing near such membranes, in particular membranes with velcro dots, membranes acting in an inclined direction, etc.; was given a complete description of the class of boundary distributions for the number of cross sections of a membrane by the discrete approximation of generalized diffusion processes that simulate the motion of a given particle (M. I. Portenko).

Were Introduced and investigated P-harmonic stationary random processes; were studied Isotropic Brownian motions, alternative to Wiener-Levy processes; were constructed Models of the Brownian motion, alternative models of Einstein-Wiener-Levy and was developed an analytical apparatus for their research; was found probabilistic solution of the hyperparabolic equation (A. F. Turbin).

Was proposed a new definition of a strong one-dimensional solution for stochastic equations; was proved the existence theorem for this solution and its connection with the weak solution was established; with the help of the criterion of weak convergence of random miraculous processes was proved the existence of an evolutionary process in which the mass is carried by independent Brownian particles (A. A. Dorogovtsev).

In 2003, for the cycle of monographs "Analytical and asymptotic methods for the study of stochastic systems and their applications" V. S. Korolyuk, A. V. Skorokhod, M. I. Portenko, A. A. Dorogovtsev and A. F. Turbin were awarded the State Prize Of Ukraine.

In algebra was found a finite-character image criterion for binovulatory partially ordered sets, were given explicit criteria for such a representation for triadic and dyadic sets, was introduced the notion of quilted quotient, were described the finite-characterized quilts, and were given the criteria for finiteness and handicapment for important classes of matrix problems (A. V. Roiter, L. A. Nazarov).

Were described the basic properties and structure of locally stepped RN groups with the condition of ball minimality, and the structures of periodic locally solvable groups that are decomposed into a product of two hyperlabel locally nilpotent subgroups, and were established the properties of groups with increasing base number (M. S. Chernikov).

In terms of the Dinkin diagrams, were proved necessary and sufficient conditions for finite-dimensional exponential and exponential growth of algebras generated by linearly connected idempotents with given spectra (Yu. S. Samoilenko).

In the theory of functions, using splines, were developed new methods for optimally restoring functional dependence on incomplete or implicit information, was solved the problem of optimization of adaptive methods for the restoration of continuous functions, were obtained precise estimates in cases where adaptive methods guarantee a higher order of error than nonadaptive (M. P. Korneichuk).

In 1994, M. P. Korneichuk was awarded the State Prize of Ukraine for the cycle of works "The theory of splines and its application in optimization of approximations".

Was obtained a further approximation-iterative method of uniform approximation of solutions of nonlinear differential and integral equations (V. K. Dzyadyk).

Developed methods that allow the only way to solve traditional problems of the approximation theory for various combining functions, in particular for well-known Weyl-Nadia and Sobolev classes and classes of functions determined by convolutions with arbitrary integrable kernels; was proposed a new approach to the classical problems of the approximation theory in abstract linear spaces (O. I. Stepanets).

Was constructed the theory of subharmonic and plurisubharmonic continuation of functions and the theory of potential for spatial capacitors, new extremal problems in the theory of univalent mappings were solved in areas that do not lie on one another, and the contour-bodily theory of thin-lomorphic and thin-hygogoho-harmonic functions were constructed without restrictions on their global weighting (P. M. Tamrazov).

In the topology were substantially developed homologous algebra and K-theory. For the space of Morse functions on surfaces, were found necessary and sufficient conditions for the membership of functions to a component of connectivity and existence criteria on a four-dimensional variety of Bott's functions with a toroidal singular set; were constructed homotopic invariants of the chain complexes of Hilbert modules over von Neumann algebras, the topological classification of functions with isolated singularities on surfaces (V. V. Sharko).

In numerical mathematics was constructed a numerical-analytic method for finding a solution to the Cauchy problem for abstract differential equations of the first and second order with an unlimited operator coefficient, which has an exponential convergence rate and allows for parallelization; were found sufficient conditions for the stability of abstract three-layer difference schemes, the coefficients of which depend on one strongly P-positive operator (V. L. Makarov).

Was constructed the theory of locally asynchronous parallel computing methods and was developed the multisite asynchronous method of investigation of nonlinear physical processes in areas of arbitrary form (B. B. Nesterenko).

In the mechanics for the nonlinear boundary value problems of the body motion motion with a liquid in the vibroacoustic field and having a free surface, were obtained the variational criteria for the stability of the interface surface and the quasi-static equilibrium form, were established new effects of overturning and failure of a limited volume of fluid; was formulated the variational principle in the nonlinear theory of motion of floating bodies partially filled with liquid; it was proved that the extremal values of the corresponding functional achieved on solutions of nonlinear boundary value problems with free boundaries describing the non-vortex motion of the external and internal fluid volume; was proposed an invariant form of nonlinear equations of perturbed motion of a rigid body with a cylindrical cavity partially filled with a liquid (I.O. Lukovsky).

Were developed the methods of studying the stability of a hard solid which revolves around its axis, stability of mechanical conservative systems; on the basis of the non-positional system of residual classes were investigated various problems of inertial navigation; were developed methods of structural decomposition and control of dynamic systems; was proposed the admissibility of the application of precession theory equations to non-stationary gyroscopic systems (V. M. Koshlyakov, S. P. Sosnitsky, S. M. Onishchenko, V. Novitsky, K. I. Naumenko).

The fundamental researches of dynamics of a solid body motion on a string suspension were carried out, for which in 1996 M. E. Temchenko and V. O. Storozhenko were awarded the State Prize of Russia.