 known only approximately, although without this possibility a theoretical solution has no practical value. The picture is more complex than it appears from the standpoint of analytic theory: the boundary value problems that can be posed correctly for different types of differential equations vary. Direct recourse to the corresponding physical concepts (such as wave propagation, heat flow, and diffusion) proves to be the most reliable guide in selecting appropriate boundary value problems for each type of equation. The related transformation of the theory of partial differential equations primarily into the theory of equations of mathematical physics has great positive significance. Works on individual types of equations of mathematical physics rightly constitute a major portion of all mathematical output. Poincare, Hadamard, Lord Rayleigh, (J. W. Strutt), Lord Kelvin (W. Thompson), Hilbert, Liapunov, V. A. Steklov, and others worked on the equations of mathematical physics after Dirichlet and Riemann.
The methods of probability theory significantly supplement the methods of differential equations in the study of nature and the solution of technical problems. In the early 19th century, probabilistic methods were mostly used in the theory of artillery fire and the theory of errors. However, at the end of the 19th century and the start of the 20th, probability theory acquired many new applications as a result of the development of statistical physics and mechanics and the elaboration of the apparatus of mathematical statistics. The most profound theoretical investigations of the general problems of probability theory in the late 19th century and early 20th were accomplished by the Russian school (Chebyshev, Markov, Liapunov).
The practical use of the results of theoretical mathematical investigation necessitates that answers to problems be given in numerical form. However, even after exhaustive theoretical analysis this often proves to be a difficult task. At the end of the 19th century and the beginning of the 20th, 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 optim