# Mathematics

**Mathematics** (математика; *matematyka*). Mathematics can be defined as the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically.

Elements of mathematical knowledge are found in the early history of Ukraine. In Kyivan Rus’ the requirements of commercial transactions, trade, taxation, land measurement, construction, architecture, military fortifications, navigation, the calendar, and other undertakings contributed not only to the development of a specific system of measurement (eg, weights and measures) but also to the application of elementary rules of arithmetic and geometry. The Rus’ way of counting was used until the introduction of the decimal system at the end of the 17th century. Some explanations of Aristotelian definitions of abstract mathematics is contained in the *Izbornik* of Sviatoslav (1073), and some mathematical problems are contained in *Ruskaia Pravda*. The first textbooks on mathematics appeared in Ukrainian in the 17th century.

One of the most distinguished professors of mathematics and physics at the Kyivan Mohyla Academy was Teofan Prokopovych. His courses in mathematics at the academy were the first in Ukraine and in the Russian Empire. Although mathematics was taught at Lviv University, it was not a subject of serious study until the second half of the 19th century.

**Beginning of the 19th century to 1917.** The development of mathematics as a separate science began in Ukraine early in the 19th century with the establishment of Kharkiv University (1805) and Kyiv University (1834). The growth of this discipline by the later 19th century is reflected in the establishment of the Kharkiv Mathematics Society (1879) and the Kyiv Physics and Mathematics Society (1890). A significant role in the development of mathematics was also played by the founding of Odesa University (1865) and Chernivtsi University (1875). Since 1897 the mathematical-natural sciences-medical section of the Shevchenko Scientific Society has published transactions that include articles on mathematics and on Ukrainian mathematical terminology.

One of the first mathematicians and physicists at Kharkiv University was Timofii Osipovsky, whose main contribution to mathematics was a three-volume treatise (1801–23), that was a basic university textbook for many years. The first Ukrainian mathematician to gain world renown was Mykhailo V. Ostrohradsky. In addition to original contributions to number theory, algebra, and geometry, he is best known for his contributions in mathematical analysis and, in particular, for his work on the transformation of multiple integrals, a theory of which he (together with G. Green and C.-F. Gauss) is regarded as the founder. The so-called Green-Ostrohradsky or Gauss-Ostrohradsky theorem forms a basis for the modern study of partial differential equations, variational calculus, theoretical mechanics, and electromagnetism. Viktor Buniakovsky is well known for his work in mathematical analysis, number theory, and probability theory. Ostrohradsky’s students I. Sokolov and E. Beyer joined the staff of Kharkiv University in 1839 and 1845 respectively. Sokolov’s teaching and research were in theoretical mechanics; Beyer was primarily interested in differential equations and the theory of probability. Beyer’s course on probability initiated a field of study for which Kharkiv University became well known. In the 1870s and 1880s most mathematics courses were taught by D. Delarue and M. Kovalsky, both graduates of Kharkiv University, and by Konstantin Andreev, V. Imshenetsky, and Matvii Tykhomandrytsky.

At the end of the 19th and the beginning of the 20th century Kharkiv University played a particularly significant role in the development of mathematics in Ukraine. During that time its faculty was joined by a group of young mathematicians that included Aleksandr Liapunov, Vladimir Steklov, A. Psheborsky, Serhii Bernshtein, and Dmytro Syntsov. They gained world recognition for fundamental results: Liapunov’s stability theory of motion and central limit theorem of probability; Steklov’s results in the theories of potential and heat conduction, the existence of Green’s function and its analytic representation, the use of eigenfunction expansion, and the notion of completeness in the solvability of boundary value problems in mathematical physics; Psheborsky’s results in the theory of elliptic functions and differential geometry; Bernshtein’s new method of determining the solvability of ordinary and partial differential equations, constructive function theory, and the axiomatic theory of probability; and Syntsov’s theory of conics and the geometry of differential equations.

At Kyiv University S. Vyzhevsky, Mykyta Diachenko, and A. Tykhomandrytsky taught mathematics from its founding in 1834. In the later part of the 19th century important research was conducted by Mykhailo Vashchenko-Zakharchenko (the use of operational calculus in determining the solvability of linear differential equations, theory of probability, history of mathematics), Vasyl Yermakov (new method of determining the solvability of the canonical system of dynamics, some problems of algebra and the convergence of series, theory of probability), Boris Bukreev (theory and applications of Fuchsian functions of rank zero, projective and non-Euclidean geometry, study of differential invariants and parameters in the theory of surfaces, history of mathematics), and P. Pokrovsky (theory of ultraelliptic functions).

At the beginning of the 20th century the mathematics faculty at Kyiv University was joined by three talented newcomers, Heorhii Pfeiffer (1900), Petro Voronets (1899), and Dmytro Grave (1902). Pfeiffer’s most important contributions were to the development of the theory of partial differential equations along the lines initiated by the famous Norwegian mathematician S. Lie. Voronets’s most important work was done in mechanics when in 1908 he derived the equation of motion for nonholonomic systems. Grave did some work in applied mathematics and mechanics, but his main interest was in algebra and number theory, where he obtained some new results in Galois theory and the theory of ideals. Several of his students (including Otto Shmidt, M. Chebotarev, B. Delone, V. Velmin, Mykhailo Kravchuk, E. Zhilinsky, and A. Ostrovsky) later became world-class algebraists. With Grave as the founder of the Kyiv algebra school, Kyiv became by 1917 the leading center of algebraic studies in Ukraine.

Mathematical research at Odesa University in the 1890s was conducted by V. Preobrazhensky (calculus, differential equations), Ivan Sleshynsky (method of least squares, mathematical logic, theory of probability), and Semen Yaroshenko (analytic geometry, least squares method). At the turn of the century the mathematics faculty was enlarged by Sleshynsky’s promising graduates I. Tymchenko (differential equations, history of mathematics, analytic functions), V. Tsimmerman (variational calculus, projective geometry), and E. Bunitsky (integral equations, ordinary differential equations, construction of Green’s functions for *n*th order equations). This group was further strengthened by the addition of the two talented mathematicians V. Kahan (axiomatic treatment of Euclidean geometry different from Hilbert axiomatics, study of Nikolai Lobachevsky geometry) and Samuil Shatunovsky (algebra, geometry, number theory, analysis, constructive approach to mathematics).

In the later 19th century, mathematical research at Lviv University was conducted by W. Żmurko (analytic geometry, algebra, and mathematical analysis) and later by J. Puzyna (analytic function theory). The basic mathematics courses at the university were taught by Puzyna and J. Rajewski. Important research activity began in 1909, when W. Sierpiński started to teach analytic number theory, theory of functions, and mathematical analysis. In 1913 Z. Janiszewski joined the faculty, and in 1917 Lviv University attracted the talented mathematician H. Steinhauss, thus initiating the famous Lviv school of functional analysis. Mathematics was also taught at the Lviv Polytechnic.

From 1893 to 1917 the mathematical-natural sciences-medical section of the Shevchenko Scientific Society published 17 volumes of its *Zapysky*. It provided a valuable forum for Ukrainian mathematicians, such as Volodymyr O. Levytsky, Mykola Chaikovsky, and Myron Zarytsky. The major emphasis of the section was to develop Ukrainian terminology in mathematics and the natural sciences. The noted Ukrainian-born mathematician Yakiv Kulyk worked at Charles University in Prague (1826–63).

**After 1917.** The structure of the All-Ukrainian Academy of Sciences (VUAN) included a division of physics and mathematics. The entire Ukrainian educational system, however, was now required to have a practical orientation. One result was that in the 1920s Dmytro Grave discontinued the algebra seminar which formed the basis of his famous prerevolutionary Kyiv school of algebra, but led a seminar in applied mathematics devoted to various technological problems. The abolition of the university system in Ukraine provided additional difficulties and spurred a wholesale departure of mathematicians from Ukraine for universities in Moscow, Leningrad, and Kazan. Scholars such as Otto Shmidt, B. Delone, and M. Chebotarev left and later became founders of different algebraic schools at their respective universities. V. Kahan left Odesa in 1923 for Moscow University, where he became the founder of a school of tensor differential geometry.

During the 1920s the VUAN in Kyiv and its affiliates—commissions on pure mathematics (chaired by Heorhii Pfeiffer), applied mathematics (Dmytro Grave), and mathematical statistics (Mykhailo Kravchuk), and the chair of mathematical physics (Mykola Krylov)—assumed the leading role in mathematical research in Soviet Ukraine. Some research was also done at the mathematical chairs of institutes of people's education, and at polytechnical, and other institutes.

In the 1930s a shift in Soviet policy led to the reorganization of research and educational institutions in Ukraine, but mathematical research continued to be harnessed to the needs of the heavy industrialization program in Ukraine. The VUAN became an association of 36 branch institutes. The Institute of Mathematics (now the Institute of Mathematics of the National Academy of Sciences of Ukraine) was established in 1934 out of three mathematical commissions and Dmytro Grave served as the institute’s first director (1934–9). After the Soviet occupation of Galicia in 1939 and Bukovyna in 1940, several distinguished Polish mathematicians of the well-known Lviv school of functional analysis (notably Stefan Banach) and some mathematicians from Chernivtsi University became affiliated with the institute. During Yurii Mytropolsky’s directorship (1958–88), the institute experienced a great expansion in research personnel and mathematical disciplines, and an improvement in the quality of research. Until the middle of 1941 the Kyiv institute was a co-ordinating center for mathematical work done at Kyiv University, Kharkiv University, Odesa University, Dnipropetrovsk University, Lviv University, and Chernivtsi University. At present, mathematical research is conducted at various institutes of the National Academy of Sciences of Ukraine, including the Institute of Mathematics of the National Academy of Sciences of Ukraine (Kyiv), Institute of Cybernetics of the National Academy of Sciences of Ukraine (Kyiv), Institute of Theoretical Physics of the National Academy of Sciences of Ukraine (Kyiv and Kharkiv), Institute of Applied Mechanics and Mathematics of the National Academy of Sciences of Ukraine (Lviv), and Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine (Donetsk). There is a mathematical section at the Physical-Technical Institute of Low Temperatures of the National Academy of Sciences of Ukraine (Kharkiv) and there are mathematics chairs at universities, technical universities, computer centers, and other institutions.

**Algebra, geometry, topology.** The development of algebra, including number theory, in Ukraine passed through two stages. The first began with the founding of the Kyiv school of algebra and number theory by Dmytro Grave in the early 20th century and ended in the late 1920s when Grave was ordered to shift his research and teaching activities from algebra (including number theory) to mechanics and various problems in applied mathematics. During the second decade of this stage Grave and his former students Otto Shmidt, B. Delone, and M. Chebotarev obtained important results in Galois theory and other areas. At the same time fundamental contributions to the theory of semigroups were made by Anton Sushkevych, the proponent of a concept known as the Sushkevych kernel. He was also the founder of the theory of quasi-groups. During the 1930s new results in algebra were obtained by Sushkevych, Mykhailo Kravchuk, and Marko Krein from Odesa.

The publication of the works of Heorhii Vorony in 1952–3 provided a fresh stimulus, and from the mid-1950s Kyiv began to regain its position as a strong center of algebraic studies. In 1955 the noted algebraist L. Kaluzhnin, who worked in the area of group theory, occupied the fledgling chair of algebra at Kyiv University. In 1956 Viktor Hlushkov joined the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR. He received world recognition for his fundamental work in topological algebras and cybernetics. In the early 1950s V. Velmin returned to Kyiv as a professor of algebra and number theory, and in 1952–4 he published his lectures on the theory of algebraic fields. In the mid-1960s an algebra section was established at the Institute of Mathematics. Its longtime director, S. Chernikov, obtained results in the study of locally solvable and locally nilpotent groups and other areas and established a useful principle of limiting solutions in the study of the algebraic theory of linear inequalities. In the early 1960s the work of Hlushkov on topological groups was continued by V. Charin, who joined Chernikov, Kaluzhnin, and the talented younger algebraists D. Zaitsev and Ya. Sysak, and later by Yu. Drozd at Kyiv University in studying the theory of groups of various types. Work in linear algebra was done by A. Roiter, L. Nazarova, and V. Bondarenko. Anton Sushkevych’s work on semigroups was continued in the 1960s by his student L. Hluskin. The chair of algebra and number theory at Kharkiv University was directed by Yu. Liubich, who led studies of periodic and almost periodic semigroups in Banach space (see Stefan Banach) and other spaces, while Yu. Drozd at Kyiv University studied mostly the representation of finite groups.

During the 1920s some important work in tensor differential geometry was initiated in Odesa, and interesting results in kinematic geometry were obtained at Dnipropetrovsk University. In Kyiv, Boris Bukreev continued his main research into various geometries and from the 1940s concentrated on aspects of Lobachevsky geometry. During that time interesting results in geometric constructions in the Lobachevsky plane were also obtained by Oleksander Smohorzhevsky of the Kyiv Polytechnical Institute. After 1961 geometry was taught at Kyiv University by M. Kovantsov, who obtained significant results in nonholonomic and projective differential geometries. Kharkiv, however, emerged as the most important center of geometry studies in Ukraine, Kharkiv University and the Physical-Technical Institute of Low Temperatures of the Academy of Sciences of the Ukrainian SSR playing a leading role in this regard. The base for a Kharkiv school of geometry was laid by Dmytro Syntsov and his students (including Ya. Blank and D. Hordiievsky). Oleksii Pohorielov, who taught at Kharkiv University from 1947, assumed the leadership of the school and turned it into a leading scientific center. V. Drinfeld, a member of that center, was awarded the Fields Medal in mathematics in 1990 for his contributions to algebraic geometry, number theory, and quantum group theory.

In the 1960s a group of Kyiv mathematicians at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR began the study of topological properties of functions and transformations, including problems in Morse theory and K-theory. Yu. Trokhymchuk has shown that if the transformation of a domain in an *n*-dimensional Euclidean space into itself is nullmeasurable and its local degree is defined and positive, except for a certain set, then it is isolated and open on the entire domain, and its local degree is positive everywhere. A series of theorems characterizing holomorphic mappings in terms of local geometric characteristics was proved by A. Bodnar in the study of multidimensional analogues of derivative operators in complex spaces. Using multivalued mappings Yu. Zilinsky obtained the geometrical criteria for strong linear convexity of compacts and domains in multidimensional complex space and solved a number of problems concerning the transformation of domains into manifolds. Vadym Sharko found necessary and sufficient conditions for the exact Morse function defined on a simply connected manifold to be isotopic, devised a method for the construction of the minimal Morse function on a non-simply-connected manifold, and developed substantially the finite-dimensional Morse theory and its applications. Quite general cover theorems and other results were obtained by P. Tamrazov. Morse functions were also studied by E. Mykhailiuk and I. Solopko.

**Theory of functions**. The most fruitful development in the theory of functions has been in the theory of approximation of functions of real and complex variables by algebraic or trigonometric polynomials or other simpler functions.

After 1917 Serhii Bernshtein continued his work at Kharkiv University in constructive function theory, for which he became internationally known in the early 1920s. In the 1920s and early 1930s Bernshtein and members of his school obtained important results in the study of absolutely monotone functions and entire functions with finite degrees. From the many members of the Kharkiv school significant contributions were made by Ya. Geronimus (orthogonal polynomials, studies of extremal properties of trigonometric and rational polynomials), V. Honcharov (approximation and interpolation of functions), B. Levin (theory of entire functions with regular growth), and B. Levitan (study of almost periodic functions defined in all of R). In 1933 Naum Akhiiezer joined the Kharkiv school and soon became its leading member. His most outstanding work consisted of deep approximation results in the constructive function theory, including the solution of the problem of Zolotarev. Outstanding results in the approximation theory were obtained in the 1950s by Volodymyr Marchenko of the Physical-Technical Institute of Low Temperatures of the Academy of Sciences of the Ukrainian SSR in Kharkiv including the study of almost periodic functions in R. Important contributions for the latter class of functions were also made in the 1930s to 1950s by O. Kovanko and I. Sokolov in Lviv, Nikolai Bogoliubov, Mykhailo Kravchuk, and S. Zukhovytsky in Kyiv, and Marko Krein and B. Korenblium in Odesa. In the mid-1930s Yevhen Remez made a significant contribution to a new aspect of the constructive function theory by developing a rigorous numerical method known as the Remez algorithm. Subsequently a similar algorithm was constructed for the rational approximation of continuous functions defined on a segment.

In the mid-1930s the works of A. Kolmogorov and J. Favard laid the foundation for the study of a new problem (know as the extremal problem on classes of functions). In 1937 Naum Akhiiezer and Marko Krein solved the extremal problem in space C for differentiable periodic functions, and somewhat later Akhiiezer solved it for analytic functions. In 1940 S. Nikolsky of Dnipropetrovsk University extended the result of Kolmogorov to other classical methods of approximation (Fejer method, interpolation polmonial, and others), to different classes of functions, and for spaces with integral and other norms. Favard proposed a conjecture known as the Favard problem, which was studied extensively in both the former Soviet Union and Europe, until it was solved completely in 1959 by Vladyslav Dziadyk. A by-product of Dziadyk’s work on aspects of the Favard problem provided the best linear approximation on classes of functions with bounded fractional derivatives. The problem of finding the exact least upper bound (l.u.b.) of the errors for the class of Hölder continuous periodic functions when the approximation is given by the linear Favard method was solved in 1961 by M. Korniichuk of Dnipropetrovsk University. In 1970 Korniichuk solved that problem in its complete generality by inventing a new method, which was subsequently used by various authors. In the late 1960s and early 1970s Korniichuk, Dziadyk, and A. Stepanets developed a new method for finding asymptotic l.u.b. when smooth functions are approximated by general Fourier summation polynomials. This method was later extended by Stepanets to the multidimensional case.

M. Korniichuk joined Vladyslav Dziadyk at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR in 1974, thereby making Kyiv a strong center for the study of the approximation theory of functions. Dziadyk and his students V. Konovalov and L. Shevchuk developed new methods for the solution of extremal problems and found the best conditions to ensure the continuation of a function in a Sobolev space from some plane set to the entire plane. Korniichuk and his school continued to work on the theory of approximation of functions, which led in 1987 to the publication of a monograph on exact constants in the theory of approximation (1987). In 1983 A. Stepanets developed a new approach in the classification of periodic functions. Yu. Melnyk developed new effective methods for the construction of entire functions with given asymptotic properties, which were then applied to the theory of system representation. V. Havryliuk found necessary and sufficient conditions for the convergence of multiple singular integrals at Lebesgue points of summable functions. These types of results in real function theory were extended by Konovalov, Shevchuk, P. Zaderei, and others. Important work in the real function theory was also carried on in Kharkiv (Yosyp Ostrovsky, Volodymyr Protsenko), Dnipropetrovsk (A. Timan), Lviv, Odesa, and other centers.

The development of the theory of functions of a complex variable was influenced by the Moscow mathematician Mikhail Lavrentev, who was appointed director of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR in Kyiv (1939–41, 1945–8) during the height of industrialization and hydrotechnical construction in Soviet Ukraine. Lavrentev made fundamental contributions to the theory of conformal and quasi-conformal mappings and its diverse application to various fields, including industrial development. His theories were further developed by a number of his students and associates in Ukraine and elsewhere. P. Bilinsky of Lviv University made significant contributions in the 1950s to the differentiability and the behavior theory of quasi-conformal mappings at isolated points and to structure theory. He was the first to introduce the basic variational method into the theory of quasi-conical maps. His students S. Krushkal and P. Biluta continued his work. In the 1960s Heorhii Suvorov of Donetsk extended a number of Lavrentev’s results for conformal and quasi-conformal mappings, including stability and differentiability theorems, to more general classes of plane and spatial transformations. Original results in the theory of functions of a complex variable were obtained in the 1950s and 1960s by Heorhii Polozhii of Kyiv, who introduced a new notion of *p*-analytic functions, defined the notions of derivative and integral for these functions, developed their calculus, obtained a generalized Cauchy formula, and devised a new approximation method for solution of problems in elasticity and filtration. His results were further developed by his students and Ivan Liashko, who solved a series of problems in the theory of filtration. P. Filchakov obtained important results in the effective approximate construction of the Riemann function, the determination of constants in the Kristoffel-Schwartz integral, and the development of a method, based on the earlier results of Lavrentev, for the solution of general problems of plane filtration in homogeneous and anisotropic grounds. Efforts of Lavrentev’s and his colleagues’ research were directed after 1945 toward the development of methods that laid the foundation for the massive hydrotechnical construction of dams, canals, and bridges on the Volga River, Dnipro River, and Don River.

Significant results in the theory of general analytical functions were obtained in the 1960s and 1970s by Ivan Danyliuk of Donetsk who used them to solve complicated PDEs and singular integral equations. Vladyslav Dziadyk obtained a simple geometrical characterization of analytic functions and their conjugates and introduced new ideas into their study. P. Tamrazov solved an important general problem concerning the limiting behavior of the holomorphic function and also solved a number of extremal problems for conformal mappings. I. Mitiuk obtained new results for univalent and multivalent conformal mappings and developed new methods for their study. Using some topological results of Yu. Reshetniak of the late 1960s, Yu. Trokhymchuk showed that the well-known theorems of D. Menshov on conformal mappings are valid in *n*-dimensional Euclidean space for *n* larger than two. Earlier V. Zmorovych of Kyiv extended significantly the Riesz-Gerglotz theorem to obtain new integral representation theorems for certain classes of univalent and nonunivalent injective holomorphic and meromorphic functions, by means of which theorems he solved many extremal problems. The approach of integral representation used widely by Kyiv mathematicians in the theory of univalent functions (1948–66) was later ‘rediscovered’ and developed anew by Western mathematicians.

Fundamental and multifaceted contributions to the theory of entire and meromorphic functions were made by scholars based in Kharkiv (Serhii Bernshtein, B. Levin, V. Honcharov, Naum Akhiiezer, Yosyp Ostrovsky, and Volodymyr Marchenko), Odesa (Marko Krein, V. Potapov), and Lviv (A. Goldberg), and earlier by Mykhailo Kravchuk and M. Chebotarev. The problem of approximating various classes of functions of a complex variable by polynomials and rational functions was considered by Mikhail Lavrentev and P. Bilinsky. But the constructive theory of functions of complex variables (similar to the one originated by Bernshtein for functions of real variables) arose from the work of Vladyslav Dziadyk in the late 1950s and in new methods he described in a series of outstanding papers in the 1960s and 1970s. Significant results were also obtained in the 1970s and 1980s by his former students L. Shevchuk, V. Bely, A. Holub, and V. Andriievsky in Donetsk and V. Konovalov, P. Tamrazov, V. Temlakov, and P. Zaderei in Kyiv.

**Functional analysis and its applications.** The development of functional analysis and its applications was a fruitful field in Ukraine, especially after 1950, when the renowned Odesa school was joined by strong functional analysis groups in Kyiv (Nikolai Bogoliubov, Ostap Parasiuk, Yurii Berezansky, Yu. Daletsky, M. Horbachuk, and H. Kats), Kharkiv (Naum Akhiiezer, Volodymyr Marchenko, Yosyp Ostrovsky, I. Glazmann, M. Kadets, Leonid Pastur, and D. Milman), Donetsk (Ihor Skrypnyk, Ivan Danyliuk), Lviv (V. Lantse, A. Plishko), and Chernivtsi (M. Popov).

The first major results in Ukraine were obtained in 1935–7 by Nikolai Bogoliubov and Mykola Krylov, who proved the existence of an invariant measure in dynamical systems that had been simply assumed until then. This proof was essential to the development of a general theory. During the first phase of development (1935–50) Marko Krein obtained fundamental and, in some cases, pioneering results in areas such as cones in Banach spaces, eigenvalue problems for positive linear operators, and topology and geometry in Banach spaces. Some of these results are now found in standard books on functional analysis as theorems of Krein, Krein-Rutman, Krein-Shmulian, Krein-Milman, Krein-Krasnoselsky, and so on. The development of functional analysis in Ukraine was enhanced after the annexation of Western Ukraine when one of the founders of its study, Stefan Banach of Lviv, joined the Ukrainian mathematical community. In the 1950s important topological and geometrical properties of Banach spaces were obtained by M. Kadets, D. Milman, and B. Levin and a series of results on extension of operators (different from Krein’s) were obtained by I. Glazmann of Kharkiv and V. Lantse of Lviv. The first essential step toward studying the spectral theory was taken in the late 1940s by Krein, who later applied his results and the theory of entire operators to the Sturm-Liouville problem on semiaxis. In 1956–65 Yurii Berezansky developed a new abstract method in the expansion theory in Hilbert spaces and used it to extend Krein’s results to partial differential operators of a certain type in functional Hilbert spaces. He also showed that an abstract self-adjoint operator in Hilbert space admits an expansion in terms of its generalized eigenvectors. Some results of the latter type were also obtained by H. Kats. In the late 1970s and early 1980s Berezansky further extended his results. The study of differential equations which can be written only in bilinear forms was undertaken by M. Horbachuk and his students in Kyiv. In the 1970s and 1980s V. Koshmanenko of Kyiv developed a general theory of dissipative operators. During this time Berezansky together with his students H. Usom, Yu. Kondriatev, and Yu. Samoilenko developed a spectral theory of elliptic operators with an infinite number of variables and applied it to some operators in quantum field theory. Further results in this direction were obtained by L. Nizhnyk for nonelliptic operators.

The first basic results on the asymptotic behavior of the spectral measure and of the spectral function for the Sturm-Liouville equation were obtained in the early 1950s by Volodymyr Marchenko, who later obtained similar results for the Schrödinger equation. The so-called inverse problem for Sturm-Liouville was solved independently by different methods in the 1950s by Marchenko, Marko Krein, and by I. Gelfand and B. Levitan in Moscow. Jointly with Yosyp Ostrovsky, Marchenko also solved the eigenvalue problem for the Hill equation with periodic boundary conditions and summarized the results in his monograph *Sturm-Liouville Operators and Applications* (1986). Yurii Berezansky was the first to study the inverse problem for equations involving partial derivatives or partial differences. In the 1960s L. Nizhnyk began detailed study of direct and inverse problems in nonstationary dissipation and, in particular, the system of Dirac equations. In 1947–8 M. Krein began the study of equations in Banach space involving bounded operator coefficients with emphasis on stability. This work was continued by Yu. Daletsky, S. Krein, and others. In 1949 Nikolai Bogoliubov and B. Khatset showed that some equilibrium problems in statistical mechanics lead to the solvability of an operator equation of the above type in a Banach space of distributions. The case of differential equations whose coefficients are unbounded operators in Hilbert space was studied by M. Horbachuk and others. The study of spectral properties, completeness of generalized eigenvectors, resolvents, and other properties of non-self-adjoint operators and operator-functions was undertaken in the 1950s to 1980s by Yu. Berezansky, M. Horbachuk, M. Krein, V. Marchenko, V. Lantse and others. In the 1950s to 1970s M. Krein developed the theory of linear operators in spaces with indefinite metric. A. Kuzhel of Kyiv dealt with spectral analysis and extensions of various classes of linear operators in the 1960s to 1980s.

Yurii Berezansky and H. Kats introduced and studied the abstract form of Sobolev spaces with positive and negative norms. The application of these spaces disclosed a series of interesting facts about partial differential equations and provided a framework for other work. Berezansky, V. Didenko, Ya. Roitberg, and others studied the Dirichlet boundary value problem for elliptic equations, smoothness of solutions, and properties of Green’s functions. Ostap Parasiuk used the generalized functions to study the solvability of integral equations. Together with Nikolai Bogoliubov he proved a fundamental theorem on the possibility of regularization of a matrix of dissipation for any order of the perturbation theory. These results were later applied to the construction of a theory of electromagnetic and weak interactions. After 1970 Ihor Skrypnyk made fundamental contributions to the development of nonlinear functional analysis, in which he introduced a new class of type (∝) nonlinear operators acting from a Banach space to its dual, developed for it a detailed topological degree theory, and applied this theory to obtain new existence theorems for a general class of abstract and concrete nonlinear partial differential equations in mechanics, elasticity, mathematical physics, and other fields. Direct methods of qualitative spectral analysis were applied by I. Glazmann in the 1960s to singular differential operators.

**Mathematical physics and nonlinear mechanics.** The first important and new results in mathematical physics in Ukraine were contained in pioneering works published in the late 1920s and early 1930s by Mykola Krylov and Nikolai Bogoliubov, who founded the renowned Krylov-Bogoliubov school. They laid the foundation of nonlinear mechanics, a new branch of mathematical physics that deals with the development of effective mathematical methods in the study of nonlinear oscillations by means of nonlinear differential equations involving a small parameter. Bogoliubov’s most important contribution, the so-called method of averaging, was made in 1945. In 1949 Yurii Mytropolsky began a sustained and systematic study of this and other asymptotic methods as well as the construction of a general theory of dynamic systems. He developed new asymptotic methods with applications to problems in contemporary physics and technology and, together with Bogoliubov, published (1961) a classic work on asymptotic methods in the theory of nonlinear oscillations. The ideas of Krylov and Bogoliubov were further developed in monographs by O. Lykova (1973), B. Moseenkov (1976), D. Martyniuk (1979), A. Molchanov (1981), H. Khoma (1983), Anatolii Samoilenko and D. Martyniuk (1985), A. Samoilenko (1987), and A. Lopatin (1988), all coauthored by Mytropolsky.

The development of mathematical physics in Ukraine after the 1930s was undertaken by Nikolai Bogoliubov, Ostap Parasiuk, V. Fushchych, Dmytro Petryna, Leonid Pastur, L. Drimfold, and others. Bogoliubov again made pioneering contributions. He was the first to provide the mathematical foundation for a consistent microscopic theory of superfluidity (1947) and constructed a mathematical theory of superconductivity (1958). The latter theory represents a fundamental achievement in theoretical physics. He also derived equations in hydrodynamics and obtained important results in quantum statistics. Together with Parasiuk he provided the mathematical foundation for the method of renormalization in quantum field theory—the so-called Bogoliubov-Parasiuk theorem. Continuing these studies, Parasiuk developed a theorem in the theory of generalized functions analogous to E. Titchmarsh’s theorem, and obtained new results in the theory of plasticity, and dynamical systems. Fushchych studied the symmetric properties of equations of mathematical physics and introduced an effective method (distinct from the method of S. Lie) for the study of symmetric properties of solutions of partial differential equations.

After the 1970s Leonid Pastur dealt with the spectral properties of various operators, an area of study which relates to mathematical physics, theoretical physics, and in particular, the theory of disordered condensed systems. In 1964 Dmytro Petryna obtained important results in the study of analytic properties of the enclosed Feynman diagram. Petryna, V. Garasimenko, and V. Malyshev studied solutions of Bogoliubov equations for an infinite three-dimensional system of particles.

**Theory of differential equations.** Mathematicians such as Serhii Bernshtein, Heorhii Pfeiffer, Mykola Krylov, Nikolai Bogoliubov, Yurii D. Sokolov, Marko Krein, Yu. Daletsky, Mikhail Lavrentev, Yosyp Shtokalo, Yaroslav B. Lopatynsky, Volodymyr Marchenko, Yurii Mytropolsky, Yurii Berezansky, Ivan Danyliuk, Ihor Skrypnyk, Anatolii Samoilenko, M. Perestiuk, Oleksander Sharkovsky, and A. Myshkis and their students made some important contributions to the theory of ordinary, functional, and partial differential equations (ODEs, FDEs, and PDEs). Fundamental applications to mechanics, elasticity, hydrodynamics, geometry, and other fields are contained in the works of Krylov, Bogoliubov, Sokolov, O. Dynnyk, Hurii Savin, Oleksii Pohorielov, Leonid Pastur, Lavrentev, and others. In addition to the well-known Kyiv school of nonlinear mechanics and the school of theoretical physics founded by Bogoliubov, the elasticity school founded in the late 1920s by O. Dynnyk in Dnipropetrovsk had a considerable influence on the development of the theory of differential equations in Ukraine, especially on the development of asymptotic methods and their application to various types of nonlinear ODEs and FDEs. As well, Yu. Sokolov obtained significant results in the theory of differential equations, which he applied to problems of analytical mechanics, and developed the Sokolov method (averaging method with functional corrections).

In 1955 Nikolai Bogoliubov and Yurii Mytropolsky published a monograph on asymptotic methods in the theory of nonlinear oscillations. It led to further refinements of the theory of asymptotic methods and to new methods of solving nonlinear ODEs and FDEs. In the 1960s the Kyiv school of nonlinear mechanics shifted its attention to at least three areas that involve certain classes of ODEs and FDEs: systems of nonlinear ODEs with impulsive action (culminating in Anatolii Samoilenko and M. Perestiuk’s monograph in 1987), ODEs with delay and/or deviating argument (A. Myshkis, Mytropolsky, D. Martyniuk, and A. Samoilenko), and FDEs and related equations (Oleksander Sharkovsky, E. Romanenko, and H. Pelekh).

After the mid-1940s the qualitative theory, particularly the stability theory of solutions of systems of linear ODEs, became a subject of considerable study. Important contributions to linear ODEs with almost-periodic and quasi-periodic coefficients were made in the 1940s and 1950s by Yosyp Shtokalo, who was the first to extend the applicability of the operational method to linear ODEs with variable coefficients. Marko Krein studied the problem of the existence and distributions of stability and instability zones for linear Hamiltonian systems with periodic coefficients, extended Aleksandr Liapunov’s results to systems of equations, and established the connection between the eigenvalues of certain differential operators and the boundary of the stability zones. For these systems Krein also obtained important results in the theory of direct and inverse spectral problems. In the 1960s M. Gavrilov and his students in Odesa obtained new and very general criteria for stability in the Liapunov sense of solutions of linear systems and even nonlinear ODEs. Interesting results concerning the asymptotic behavior of solutions of linear ODEs and systems of ODEs were obtained in the 1950 and 1960s by Stepan Feshchenko, M. Shkil, and I. Rapoport under various conditions on the coefficients.

In the 1960s Vitalii Skorobohatko and E. Bobyk in Lviv found the necessary and sufficient conditions for the solvability of the Vallée-Poussin problem for linear ODEs of the *n*th order in an arbitrary interval, and the analogous sufficient condition for the solvability of nonlinear ODEs of any order. Fundamental results for direct and inverse spectral problems for one-dimensional Sturm-Liouville equations were obtained by Volodymyr Marchenko. Qualitative studies of and spectral analysis of singular differential operators were done by I. Glazmann in the 1960s.

Although his work at Kharkiv University after 1917 dealt mostly with the theory of probability and constructive function theory, Serhii Bernshtein also made qualitative studies of some PDEs. In Kyiv Heorhii Pfeiffer continued his studies of linear systems of PDEs of the first order with one unknown function. While in Soviet Ukraine during the 1940s, Mikhail Lavrentev developed the theory of quasi-conformal mappings that determined a new geometric approach to the theory of PDEs with applications to hydrodynamics, filtration detonation, and other fields. Some of his results in the theory of filtration were extended by P. Filchakov and his students. The foundation of the general theory of boundary value problems for linear systems of PDEs of elliptic type is contained in the works of Yaroslav B. Lopatynsky. In addition to making other important contributions to the theory of linear PDEs, he was the first to identify the condition for the compatibility of the coefficients of the elliptic system with the coefficients of the boundary operator, now known as the Lopatynsky condition. In the 1950s Yurii Berezansky was the first to develop a method for dealing with the inverse problem of spectral analysis for PDEs. Among other results, he developed the spectral theory for self-adjoint PDEs in unbounded domains. L. Nizhnyk studied the nature of the spectrum for nonelliptic PDEs. In addition to his work on systems of ODEs with delay in the 1960 and 1970s, A. Myshkis obtained interesting results for PDEs of hyperbolic and other types. In his work on PDEs Vitalii Skorobohatko obtained a generalization of the Gerlotz formula, solved the Cauchy problem in case of multiple roots of the characteristic equation, and made a deep study of the solvability of the Dirichlet problem for systems of elliptic PDEs. In the 1970s and 1980s the asymptotic methods of nonlinear mechanics of Mykola Krylov, Nikolai Bogoliubov, and Yurii Mytropolsky were extended to PDEs in monographs by B. Moseenkov, H. Khoma, A. Bakai, and Yu. Samoilenko, all written jointly with Mytropolsky. The celebrated Stefan problem was studied in the 1980s by several researchers from Donetsk, including Ivan Danyliuk. The Kordeweg-de Vries equation was solved by Volodymyr Marchenko in 1972 and the Cauchy problem for the Kordeweg-de Vries equation was studied in the 1980s by V. Kotliarov and Ye. Khruslov.

After the 1960s Ivan Danyliuk made significant contributions to the theory of PDEs and nonlinear problems in mathematical physics with free (unknown) boundaries. He applied his results to various problems of physics (such as the Stefan problem), mechanics, and other fields. After 1970 deep and systematic studies of abstract and concrete very general nonlinear elliptic PDEs of the higher order of divergence type were conducted by Ihor Skrypnyk, who developed and worked with a new topological degree theory for the abstract nonlinear operator of type (∝) from a separable Banach space to its dual. He extended his results to some nonelliptic PDEs and to elliptic equations which are not of divergence form.

**Probability and statistics.** Until 1917 the theory of probability and mathematical statistics was developed in Ukraine by mathematicians in Kharkiv (Andrii Pavlovsky, Matvii Tykhomandrytsky, Aleksandr Liapunov, and Serhii Bernshtein), Kyiv (Mykhailo Vashchenko-Zakharchenko, Vasyl Yermakov, and Yevhen Slutsky), and Odesa (Ivan Sleshynsky). The development of the theory in Ukraine after 1917 can be divided into two phases, 1917–45 and 1945 to the present. The first phase focused mainly on such related fields as limit theorems in probability, the theory of random processes, and mathematical statistics and has involved such mathematicians as Mykhailo Kravchuk, Slutsky, Bernshtein, Mykola Krylov, Nikolai Bogoliubov, and Y. Hikhman. Kravchuk studied orthogonal polynomials corresponding to discrete probabilistic distributions (now known as Kravchuk’s polynomials). Slutsky obtained a series of interesting results in the theory of random functions which contributed to the development of the theory of random processes. Bernshtein extended Liapunov’s central limit theorem, made the first attempt at the axiomatic construction of the theory of probability, initiated the study of stochastic equations, and considered a very special case of the Markov process. A different approach to the qualitative study of random processes was proposed by Krylov and Bogoliubov. Hikhman developed the theory of stochastic differential equations.

The second phase of development started in 1945, when Borys Hniedenko joined the Lviv branch of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR and immediately organized a section on the theory of probability and mathematical statistics. He wrote a textbook in Ukrainian on probability theory and coauthored a monograph on the boundary resolution for sums of independent random quantities (1949), which explained his ideas on local limit theorems in probability. The latter work had a substantial impact both within and outside the USSR. Further results on local limit theorems were obtained by Ostap Parasiuk and Kateryna Yushchenko, who used them to solve problems in statistical mechanics and physics.

In 1949 Borys Hniedenko joined the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR in Kyiv and organized a section on the theory of probability and mathematical statistics. It studied nonparametric problems in mathematical statistics and the theory of massive service. The section attracted talented young mathematicians such as Volodymyr Koroliuk, Anatolii Skorokhod, M. Portenko, A. Husak, A. Dorogovtsev, V. Buldygin, I. Yezhov, A. Turbin, H. Butsan, Volodymyr Mykhalevych, and N. Slobodeniuk and constituted the nucleus for the so-called Kyiv school of probability theory. A. Skorokhod’s pioneering monograph on random processes and stochastic differential equations (1961) was followed by new results on the martingal problem and equations in infinite-dimensional space. He introduced a number of new notions and methods now called Skorokhod space, Skorokhod topology, Skorokhod versions of weak convergence, the Skorokhod convergence theorem, and the Skorokhod-Wichura-Dudley theorem. Solutions of stochastic differential equations were obtained by Portenko; problems in the stability of solutions of differential with random coefficients were studied by Yurii Mytropolsky, Anatolii Samoilenko, and Y. Hikhman. In the early 1970s Hikhman and Skorokhod published a three-volume treatise on the theory of stochastic processes. Skorokhod obtained original results in studying various aspects of the Markov processes, one of the main areas of study in the probability and statistics section. He was later joined by younger specialists. Portenko and R. Boiko studied branching processes, while V. Koroliuk and A. Husak worked on boundary problems for random processes.

Semi-Markov processes began to be studied in the USSR in the mid-1960s. In 1965 Volodymyr Koroliuk solved a problem that was a key to various application problems. He also studied the asymptotic analysis of the distribution of functionals related to semi-Markov processes. Together with A. Turbin he obtained new results for larger classes of Markov and semi-Markov processes and applied them to some problems of statistical physics and hydrodynamics.

In the 1970s Anatolii Skorokhod developed further the theory of probabilistic measures in infinite-dimensional spaces and established the theory of quasi-invariant measures in Hilbert spaces, which provided the most general conditions for the absolute continuity of measures under nonlinear transformation. Important studies in this area were done also by V. Buldygin and H. Syta.

The study of evolutionary random families represents a new direction in the theory of random processes founded earlier by the members of the institute. In the 1960s Anatolii Skorokhod suggested a new method for describing matrix noncommutative random processes with independent multiplicative increments and obtained new results by means of classical random processes. These studies were continued by H. Butsan, who introduced a general notion of stochastic subgroups as a random two-parameter family of operators satisfying the evolutionary relation and indicated important classes of stochastic continuous multiplicative semigroups. Further studies in this and other areas were done by Skorokhod, Volodymyr Koroliuk, and their former students. In the 1980s the Kyiv school of probability theory made rapid progress in its work on the theory of probability and mathematical statistics: its members, such as V. Buldygin, V. Girko, D. Silvestrov, Yu. Daletsky, A. Volpert, A. Husak, and Butsan, published dozens of monographs, many of which were translated into English. It consolidated its international reputation as a leading center of mathematical research.

**Approximation methods for solving abstract and differential equations.** Starting in the 1920s Mykola Krylov, Nikolai Bogoliubov, and Mykhailo Kravchuk developed new methods for obtaining fundamental results in the areas of approximate solvability of differential equations and abstract operator equations. Krylov first provided a rigorous justification for the use of variational methods and the direct methods of Ritz and least squares for the approximate solvability of self-adjoint problems of mathematical physics. Krylov (later with Bogoliubov) succeeded in obtaining effective error estimates for his approximation methods. Kravchuk studied the existence and convergence of the approximate solutions obtained in Krylov’s research when applied to ODEs, PDEs, and integral equations. He obtained good error estimates and characterized the speed of convergence depending on the coefficients of the equations.

In the 1950s M. Polsky in Kyiv made a deep study of the abstract projection method (mostly in Hilbert spaces), which included methods of the Ritz-Galerkin type. A number of results for Ritz-Galerkin and projection type methods, dealing mostly with the study of good and effective error estimates, the speed of convergence, and the stability of the methods, were obtained by A. Luchka in Kyiv in the late 1960s. From the late 1950s A. Martyniuk in Zhytomyr studied the approximate solvability of odd ODEs, complicated PDEs, and abstract operators in a Hilbert space.

At the end of the 1950s Yurii D. Sokolov introduced and studied a new and effective method for the approximate solvability of differential and integral equations. Further development of the Sokolov or the averaging method with functional correction was done by A. Luchka and N. Kurpel. In the late 1950s Heorhii Polozhii began an intensive effort to construct an effective approximate method for solving boundary value problems in mathematical physics, which led to the development of the method of summary representation.

Vladyslav Dziadyk made a major contribution to the theory of approximation methods for solving operator equations. He examined ways of using the Chebychev theory of the approximation of functions in order to construct new and practically effective methods for the solution of differential and integral equations. His results, as well as those obtained by his students in the 1970s and 1980s, were summarized by Dziadyk in a 1988 monograph on approximation methods for the solution of differential and integral equations.

**Mathematicians in the West.** Important results have been obtained by mathematicians of Ukrainian ancestry in the West, most of whom worked or are working as professors at various universities in the United States, Canada, Australia, and other countries. They include Osyp Andrushkiv, R. Andrushkiv (eigenvalue problems for linear and nonlinear K-symmetric operators, solution of nonlinear parabolic PDEs with application to cryosurgery, theory of hydrodynamics stability), I. Bohachevsky (applied mathematics, stochastic optimization), Mykola Derzhko (operator theory, PDEs, applied mathematics, operational research), I. Hawryshkewycz (computer sciences, development of new areas of research in computer sciences, structure techniques for analysis of information systems), B. Lawruk (general theory of PDEs, special symplectic spaces, symplectic relations to PDEs, involutory conditions for Riemann invariants), W. Madych (applied mathematics, including certain aspects of signal processing, computerized tomography, classical Fourier theory and approximations), A. Nagurney (applied mathematics, operational research), A. Ostrowski (algebra, geometry, analysis), Wolodymyr Petryshyn (iterative and projection methods, fixed point theorems, nonlinear Friedrichs extension, A-proper mapping theory, solvability of ODEs and PDEs, topological degree for multivalued K-set-contractions [with P. Fitzpatrick]), Liubomyr Romankiv (computer sciences, operational research), E. Seneta (theory of probability and mathematical statistics, history of mathematical statistics), M. Skalsky (theory of probability, statistics, applications), W. Vasilaki (new probabilistic approach to the approximation methods for PDEs and boundary layer problems, computerized tomography, nuclear magnetic resonance), and R. Voronka (analysis, applied mathematics, development of effective methods for teaching mathematics).

**History of mathematics.** Until 1917 the study of the history of mathematics in Ukraine focused mostly on the history of a specific field, institution, or mathematician. Historical research was conducted sporadically by mathematicians in Kharkiv (Andrii Pavlovsky, Dmytro Syntsov), Kyiv (Mykyta Diachenko, Mykhailo Vashchenko-Zakharchenko, Boris Bukreev, Dmytro Grave), and Odesa (I. Tymchenko, V. Kahan, Samuil Shatunovsky). In 1917–50 the research became more systematic: Grave studied the history of algebraic analysis; Mykola Krylov, the emergence of variational calculus and the role of the minimum principle in contemporary mathematics; Mykhailo Kravchuk, mathematics and mathematicians at Kyiv University in 1835–1935, the influence of Euler on the development of mathematics; M. Marchevsky, history and development of mathematics chairs and the first 75 years of the Kharkiv Mathematics Society; Anton Sushkevych, mathematics at Kharkiv University in 1805–1917 and the history of group theory; Syntsov, detailed study of the works of the faculty of Kharkiv University; Tymchenko, history of logarithms; D. Kryzhanivsky, lectures on the history of mathematics given at Odesa University; Volodymyr O. Levytsky, mathematics at the Shevchenko Scientific Society and the Lviv (Underground) Ukrainian University; Mykola Chaikovsky, bibliography of Ukrainian mathematics; and Myron Zarytsky, historical developments in mathematics.

After the 1950s the history of mathematics gained increasing attention at Ukrainian universities and the Academy of Sciences of the Ukrainian SSR in Kyiv. In 1956 the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR established a separate section on the history of mathematics under the supervision of Yosyp Shtokalo and a special seminar, which quickly became republican in scope. Between 1956 and 1964 the section published 13 volumes, in Ukrainian, on the history of mathematics as well as some special historical books. It also published the collected works of distinguished Ukrainian mathematicians, including Mykhailo V. Ostrohradsky (3 vols, 1959–61), Mykola Krylov (3 vols, 1949–61), Heorhii Vorony (3 vols, 1952–3), Nikolai Bogoliubov (3 vols, 1964), Dmytro Grave (1971), and Aleksandr Liapunov (1982). L. Hratsianska and V. Dobrovolsky have written a series of papers on the development of mathematics at Kyiv University and biographies of Kyiv mathematicians such as Mykhailo Vashchenko-Zakharchenko, Vasyl Yermakov, Dmytro Grave, Boris Bukreev, and Heorhii Pfeiffer. B. Bily studied the contributions of Aleksandr Liapunov, Vladimir Steklov, and S. Kovalevska, while I. Naumov published a biography of Dmytro Syntsov and Naum Akhiiezer wrote a book about Serhii Bernshtein. A. Bogoliubov wrote a collection of short biographies of Kyiv mathematicians (1979) and *Matematiki i mekhaniki* (Mathematicians and Mechanicians, 1983). Book-length biographies of S. Kovalevska (1955), Grave (1940, 1968), Ostrohradsky (1963), Mykhailo Kravchuk (1979), and Krylov (1987) appeared.

In the 1950s Y. Hikhman and Borys Hniedenko examined the development of the theory of probability in Ukraine. Hniedenko and Y. Pohrebysky published several articles on the development of mathematics in Ukraine. Ya. Blank, D. Hordiievsky, and Oleksii Pohorielov prepared a history of geometry at Kharkiv University (1956). Yosyp Shtokalo edited a Russian-Ukrainian mathematical dictionary (1960) and a Ukrainian mathematical bibliography for 1917–60 (1963). In the late 1950s and in the 1960s Marko Krein, S. Kiro, and others wrote on the history of mathematics in Odesa. Mykola Chaikovsky wrote on the history of mathematics in Lviv. Some historical studies on nonlinear mechanics by Yurii Mytropolsky appeared in the 1970s. M. Pavlenko wrote a study of Viktor Hlushkov’s ideas (1988).

Although some attempts to write a separate history of mathematics in Ukraine have been made, the most informative source for the subject is the four-volume *Istoriia otechestvennoi matematiki* (A History of the Fatherland’s Mathematics, 1966–70), edited by Yosyp Shtokalo, which deals with the development of mathematics in the countries of the former USSR.

Wolodymyr Petryshyn

[This article originally appeared in the *Encyclopedia of Ukraine*, vol. 3 (1993).]