h, the numerical methods of analysis grew into an independent branch of mathematics. Here, particular attention was paid to the methods of numerical integration of differential equations (the methods of Adams, Stormer, Runge, and others) and to quadrature formulas (Chebyshev, Markov, Steklov). The extensive development of works requiring numerical calculations led to the necessary compilation and publication of an ever-growing number of mathematical tables.
Intensive work on the problems of the history of mathematics was begun in the second half of the 19th century.
Based on A. N. KOLMOGOROV’S article in the 2nd edition of the Bol’shaia Sovetskaia Entsiklopediia
CONCLUSION. The foregoing discussed the main features of modern mathematics and listed the main directions of mathematical research according to branches as they had formed by the early 20th century. This division into branches largely still exists despite the rapid development of mathematics in the 20th century, especially since World War II. (The current state of mathematics and the achievements of various schools and individuals are reflected in the corresponding articles on number theory, algebra, logic, geometry, topology, function theory, functional analysis, differential equations, equations of mathematical physics, probability theory, mathematical statistics, and computer science.)
The requirements of the development of mathematics itself, the “mathematization” of various fields of science, the penetration of mathematical methods into many spheres of practical endeavor, and the rapid progress of computer technology have led mathematicians to redirect their efforts within the various branches of mathematics and to develop new mathematical disciplines, such as the theory of algorithms, information theory, game theory, operations research, and cybernetics.
Discrete, or finite, mathematics has arisen on the basis of the problems of the theory of control systems, combinatorial analysis, the theory of graphs, and coding theory.
The problems of the optimal (in various senses of the word) control of physical or mechanical systems, which are described by differential equations, have led to the creation of the mathematical theory of optimal control. The related problems of the control of objects in conflict situations have resulted in the creation and development of the theory of differential games.
Studies in control theory and in related fields, combined with the progress in computing technology, have provided the foundations for automating new spheres of human activity.
Soviet mathematics holds a leading place in world mathematical science. The works of Soviet scientists play a decisive role in many fields. The advances of prerevolutionary Russian mathematics were connected with the work of outstanding individuals and encompassed a narrow range of topics. There were scientific mathematical centers in only a few cities (St. Petersburg, Moscow, Kazan, Kharkov, Kiev). The main achievements were associated with the work of the St. Petersburg school. After the Great October Socialist Revolution a number of important new directions were followed in the Moscow mathematical school. Such universities as St. Petersburg, Moscow, and Kazan were the main centers of mathematical research in prerevolutionary Russia. Since 1917 the development of mathematical research and applied mathematics has been closely linked with the development and growth of the Academy of Sciences of the USSR. Studies in mathematics are largely concentrated in the institutes of mathematics of the Academy of Sciences of the USSR, the academies of sciences of the Union republics, and leading universities. During the Soviet era the rise of many scientific schools in cities where previously no significant work was done in mathematics has been an important feature of the mathematical development in the Soviet Union. Such are the mathematical schools in Tbilisi, Yerevan, Baku, Vilnius, Tashkent, Minsk, Sverdlovsk, and other cities and the scientific school that was created in the 1960’s at Akademgorodok, nea