## Development of the institute in the period 1960 - 1989

In 1958 - 1988, the Director of the Institute of Mathematics was Yu. O. Mitropolsky, and since 1988 the institute has been headed by A.M. Samoilenko.

At the beginning of this period, the Institute operated 9 departments and 2 laboratories: the Department of Mathematical Physics (Yu.O. Metropolsky), differential equations (Yu. D. Sokolov), functional analysis (O. S. Parasyuk), mathematical analysis (Yu. M. Berezansky), probability theory and mathematical statistics (B. V. Gnedenko), general mechanics (O. Yu. Ishlinsky), mathematical theory of elasticity (G. M. Savin), history of mathematics (Y. Z. Shtokalo), Kharkiv department on geometry (A. V. Pogorelov), the laboratory of computational mathematics and technology (V. M. Glushkov) and higher nervous activities (M. M. Amosov).

Researches at that time were conducted in the following scientific areas: the theory of nonlinear oscillations and mathematical physics, the theory of differential equations, probability theory and mathematical statistics, functional analysis, the theory of functions, topology, algebra, the dynamics of special mechanical systems, geometry, computational mathematics and history of mathematics.

In 1962 for research in the field of mechanics M. M. Sity was awarded the Lenin Prize.

In the theory of differential equations and the theory of nonlinear oscillations were established the general laws of construction of asymptotic methods of nonlinear mechanics, was developed mathematical theory of multifrequency oscillations; continued further development of the method of averaging; Asymptotic methods were extended to new classes of equations with partial derivatives, to equations with delay, and others; was developed the method of asymptotic splitting of differential systems on the basis of the group-theoretical approach; continued further development of the method of integral manifolds and the method of successive substitutions with accelerated convergence of iterations (Yu. O. Mitropol'skii, A. M. Samoilenko, O. B. Likova, V. I. Fodchuk).

In 1965, for a series of works devoted to the essential development and strict mathematical substantiation of the theory of nonlinear oscillations, Yu. O. Mitropolsky was awarded the Lenin Prize.

In 1970, for the development of electrical integrators EGDA and the introduction into practice of simulation of the methods of electrohydrodynamic analogy, P.F. Filchakov and V. I. Punchyshyn were awarded the State Prize of the Ukrainian SSR.

During this period, the theory of subtraction in the quantum field theory and the method of renormalization of the quantum field theory were modified (M. M. Bogolyubov, O. S. Parasyuk), also was completely solved the problem of regularization of divergent integrals.

It was proved the theorem about the impossibility of constructing an axiomatic quantum field theory with a positive spectrum of the energy-momentum operator, and established the criterion of holomorphy of the scattering amplitudes by energy and transmitted momentum (D. Ya. Petrina).

Important results were obtained in the theory of dynamical systems and structural stability (O. M. Sharkovsky).

It was developed a new method for studying the group properties of differential equations and constructed a multiparametric families of exact solutions of multidimensional nonlinear equations of mathematical physics (V.I. Fushchich).

Variational methods of solving the basic boundary value problems of a limited volume of ideal fluid were proposed. Properties of the spectrum and eigenfunctions of the problem on the eigenvalues with a parameter in the boundary conditions, which describes free fluctuations of the fluid in the capacities of arbitrary geometric shapes were investigated (S. Feshchenko, I. O. Lukovsky, M. Ya. Barnyak, O. N. Komarenko).

It was created a new promising direction - the asymptotic phase consolidation of random processes focused on the study of the evolution of complex stochastic systems. Has been developed the theory of service and reliability theory, the method of factorization in boundary problems for processes with independent increments, a number of limit theorems for semi-Markov processes have been proved (V. S. Korolyuk, M. S. Bratiychuk, I. I. Ezhov, A. F. Turbin , D. V. Gusak).

In the theory of multiplicative stochastic semigroups a general theory of random operators was constructed, significant results were obtained in the theory of stochastic differential equations with generalized transport coefficients, operator stochastic equations and random series in infinite-dimensional spaces were investigated, the general ergodic theorem for Markov processes was proved (A. V. Skorokhod, M. I. Portenko, V. M. Shurenkov, V. V. Buldygin, G. P. Butsan).

New results were obtained in the theory of linear inequalities allowed solving many problems of optimization, economics and pattern recognition (S. M. Chernikov).

In the terms of differential graded categories, a general theory of matrix problems was constructed, an important problem of Bauer-Trelles (A. V. Roiter, L. O. Nazarova) was solved.

During this period, the theory of decompositions was constructed based on general generalized vectors of the general families of commuting normal operators and the theory of generalized functions of an infinite number of variables, the direct and inverse problem of nonstationary scattering was solved for hyperbolic systems and transfer equations, the spectral theory of boundary value problems for the differential - operator equations was developed, the operator approach to the theory of boundary values of solutions of partial differential equations in various spaces of ordinary and generalized functions was proposed and applied (Yu. M. Berezansky, L. P. Nyzhnik, M. L. Gorbachuk).

Asymptotic equations for the upper bounds of deviations of multiple Fourier sums in the classes of continuous periodic functions of several variables were obtained in the theory of functions and their approximations, were developed effective methods for investigating extreme problems in the approximation theory, which in many cases of the approximation of functions by polynomials and splines made possible the obtaining of completed results, in particular, in problems of optimal restoration of functions and linear functionals (V.K. Dzyadyk, M.P. Korneichuk, A.I. Stepanets).

In 1978 M.P. Korneichuk was awarded the State Prize of the USSR for the development of effective methods of the approximation theory.

Significant results were obtained in the topology and complex analysis: for a wide class of fundamental groups the existence of exact Morse functions on manifolds was proved and a series of results in a stable algebra were established (the existence of minimal resolvent, chain complexes, etc.); the geometric criteria of strong linear convexity of compacta and regions in a multidimensional complex space were found with the help of multivalued mappings and a number of problems concerning the display of regions on manifolds were solved; the important extremal problems of the theory of conformal mappings, in particular, the known extreme problem about the capacitance capacitor were solved (V. V. Sharko, A. V. Bondar, Yu. B. Zelinsky, P. M. Tamrazov).

Strong results were obtained about the solid state motion around a fixed point, were developed effective algorithms for estimating accuracy and optimal control for inertial navigation systems (V. M. Koshlyakov).

In 1976 V. M. Koshlyakov was awarded the State Prize of the USSR for work on the theory of gyros.

A number of problems were solved in the theory of control which were arising during the creation of robotic systems, in particular the task of stabilizing the walking apparatus (V. B. Larin).

An important researches were performed in the field of nonlinear mechanics of a solid with liquid-filled cavities; a new approach to the analysis of the stability of systems of linear differential equations with random coefficients was developed (I. O. Lukovsky, V. A. Trotsenko, D. G. Korenivsky).

Significant results were obtained about the problems of mechanics at the Lviv Branch of the Institute of Mathematics. The cycle of research on the solution of applied thermoelasticity problems in shell designs, performed by Ya. S. Pidstryhach, Y. I. Burak, G. V. Plyatsky and B. I. Kolodiy, was awarded the State Prize of the Ukrainian SSR in 1975.

In 1978, was created the Institute of Applied Problems of Mechanics and Mathematics on the basis of the Lviv Branch of the Institute of Mathematics.

For the work on creation of the Encyclopedia of Cybernetics, performed in close cooperation with the scientists of the Institute of Cybernetics, in 1978 V. S. Korolyuk was awarded the State Prize of the Ukrainian SSR.

At the end of this period (1980 - 1989), the scientific and scientific-organizational activities of the Institute were aimed at the further development of mathematical science, increasing the efficiency of its use in applied purposes, and ensuring the first-rate development of fundamental research in the following priority areas: asymptotic and qualitative methods in the theory of differential equations, analytic methods of the theory of random processes, functional analysis, theory of approximation of functions, dynamics and stability of special multidimensional systems.

In 1980 Yu. O. Mitropolsky, V. M. Kalinovich and V. B. Larin were awarded the State Prize of the Ukrainian SSRF for applied developments in the field of nonlinear oscillation theory.

At this period, the theory of perturbations of invariant toroidal manifolds of dynamical systems and the basis of the theory of pulsed systems were developed, metric theorems of convexity of linear systems with quasiperiodic coefficients were proved, the abstract principle of the construction in the theory of stability was formulated and revised. In 1985, for these works A. M. Samoilenko was awarded the State Prize of the Ukrainian SSR.

A constructive theory of the Euclidean scattering matrix was created on the basis of equations for coefficients, for certain models was proved a theorem for the existence of solutions (D. Ya. Petrina, A. L. Rebenko, V. I. Skrypnik).

The states of infinitely equilibrium classical systems within the framework of the formalism of the canonical ensemble were constructed (M. M. Bogolyubov, D. Ya. Petrin).

A significant contribution was made to the development of a constructive method for studying the symmetric properties of multidimensional systems of partial differential equations; described systems of linear and nonlinear differential equations invariant with respect to Galilean, Poincare and conformal groups; was constructed wide classes of exact solutions of many-dimensional nonlinear wave levels (V. I. Fushchych).

The basis of a qualitative theory of functional-difference equations with a continuous argument was constructed. A new approach in mathematical modeling of turbulence was proposed, which made it possible to explain such phenomena as self-modality, autostochasticity, etc. (O. M. Sharkovsky).

An original concept of the development of parallel computing for the study of nonlinear physical processes in free form fields which are described by the equations of mathematical physics was developed (B. B. Nesterenko).

For solving integral, differential, integro-differential equations projective-iterative methods with high convergence velocity were developed (A. Yu. Luchka).

Approximate methods of solving nonlinear problems of heat conduction and diffusion, ecological problems with free boundaries were developed (A. A. Berezovsky).

The theory of stochastic differential equations in spaces with complex local structure (varieties with edges, varieties with variable number of measurements, etc.) and the theory of linear stochastic differential equations in infinite-dimensional spaces with coefficients that are unbounded linear operators were constructed; limit theorems for an infinitely increasing number of interacting particles were proved; it were given a description of the class of boundary random processes (A. V. Skorokhod, M. I. Portenko) and a complete description of the finite probabilities of Markov's ergodic processes with a general phase space and probability measure carriers in Banach spaces (VM Shurenkov, VV Buldygin). For evolutionary stochastic operator systems, theorems on the isomorphism of such systems for ordinary random processes with independent increments were proved(G. P. Butsan). With the help of direct probabilistic methods, formulas of duality for random walks were obtained (I.I. Ezhov).

In 1982 A.V. Skorokhod was awarded the State Prize of the Ukrainian SSR for a study on general theory and special classes of random processes.

At this period the theorems of the type of asymptotic phase consolidation for semi-Markov random evolutions were proved, the boundary behavior of such evolutions and additive functionals in the phase consolidation scheme and the asymptotic behavior of solutions of systems of differential equations with coefficients depending on Markov processes were studied, a new analytical direction in the mathematical theory of reliability of complex restorative systems was developed (V. S. Korolyuk, A. F. Turbin).

In the field of functional analysis, has been developed further the spectral theory of self-directed and normal operators operating in the spaces of functions of infinite number of variables and were established new features of self-inflection of infinite-dimensional elliptic operators (Yu. M. Berezansky); was constructed the theory of extensions of Hermite operators of boundary values (A. N. Kochubei) and its applications to nonclassical differential operators (point interactions, strongly singular potentials, etc.)(A. N. Kochubei, V. A. Mikhailets, L. P. Nyzhnik ); was developed the theory of smooth and generalized functions (vectors), based on which an arbitrary closed linear operator in a Banach space is placed in lieu of the differentiation operator in the space of functions suited with a square of squares; For spaces of such functions was proved an abstract variant of the Paley-Wierer and Stone-Weierstrass theorems (M. L. Gorbachuk, V. I. Gorbachuk). For the self-adjoint operators generated by an elliptic differential expression and arbitrary boundary condition was investigated the structure of the spectrum, classes of boundary conditions, in which the spectrum is discrete, was studied the asymptotic behavior of this spectrum (V. I. Gorbachuk, V. A. Mikhailets); were proved the theory of scattering is constructed in terms of bilinear functionals and the methods for investigating singular perturbations of self-directed operators; also was proved the existence of wave operators in a number of models of quantum field theory (V. D. Koshmanenko, L. P. Nyzhnik).

The signs were found of the equivalence of the root vectors of polynomial operators of operators, and the theorems on the minimality and basisness of root vectors are obtained (G.V. Radziyevsky).

Have been Investigated multivariate inverse scattering problems for hyperbolic equations with partial differences, integro-differential and functional equations; nave been Integrated space-two-dimensional nonlinear evolution equations by the inverse scattering problem (L.P. Nyzhnik). In 1987, for the practical application of the results L.P. Nyzhnyk was awarded the State Prize of the Ukrainian SSR.

In the theory of functions were obtained important results concerning the problem of polynomial approximation and spline approximation, and solved extremal problems of approximation of certain classes of functions and the problem of optimal coding and optimal restoration of functions and linear functionals (M. P. Korneichuk).

An approximation-iterative method for ordinary differential equations with an analytic right-hand side was developed and a generalized problem of moments was investigated (V. K. Dzyadyk).

The foundations of the approximation theory were established on the classes of periodic functions given by means of multipliers and displacements of the argument (O. I. Stepanets).

Significant results concerning the problem of extension of functions from Sobolev's space were obtained by V. M. Konovalov and I. O. Shevchuk.

Global and local contour-solid theorems for holomorphic functions and maps in open sets of a closed complex plane were proved (P.M. Tamrazov).

Have been achieved significant results in the study of topological properties of functions and mappings, in particular, in Morse theory and K-theory (Yu. Yu. Trokhimchuk, A. V. Bondar, Yu. B. Zelinsky, V. V. Sharko).

Fundamental results were obtained in the theory of groups and linear algebras, in particular, were studied and constructed an important types of non-invariant periodic groups with Abelian commutator and Abelian Sylow subgroups (S. M. Chernikov, D. I. Zaitsev), was indicated the theory of representations of generalized partially ordered sets and its important applications to finite-dimensional algebras (L.O. Nazarova, A.V. Roiter).

Significant progress has been made in solving complex mathematical problems of mechanics, in particular has been further developed the theory of gyroscopes and navigational gyroscopic systems (V.M. Koshlyakov).

The study of the dynamics of fluid motion, performed by M. Y. Temchenko, in 1981 was awarded the State Prize of the USSR.

In 1983, for scientific work in the field of dynamics of special mechanical systems, the creation of new mathematical models of mechanics of solid deformed bodies, and the development of methods for calculating the oscillations and the stability of such bodies I. O. Lukovsky, D. G. Korenivsky, M. O. Pustovoitov, V. A. Trotsenko were awarded the State Prize of the Ukrainian SSR.