ems having a finite number of degrees of freedom) and partial differential equations (in the study of continuous media and in quantum physics). Therefore, all directions of investigations of differential equations were intensively developed during this period. The methods of operational calculus were created to solve complex linear systems. The method of parametric expansion was used widely in the study of nonlinear systems having small nonlinearity. The analytic theory of ordinary differential equations continued to be developed (Poincare and others). However, in the theory of ordinary differential equations, problems of the qualitative investigation of their solutions now attracted the greatest attention: the classification of singular points (Poincare and others) and problems of stability, which were studied in great depth by A. M. Liapunov.
The qualitative theory of differential equations served for Poincaré as a starting point for a broadly conceived continuation of the investigations, barely outlined by Riemann, into the topology of manifolds, particularly the study of the fixed points of continuous mappings of manifolds onto themselves. The “combinatorial,” “homologous,” and “homotopic” methods of modern topology originated here. Another direction arose in topology based on set theory and functional analysis and led to the systematic development of the theory of general topological spaces.
The theory of partial differential equations assumed an essentially new form at the end of the 19th century owing to intense interest in boundary value problems and the elimination of the restriction to analytic boundary conditions. The analytic theory, which originated with Cauchy, Weierstrass, and S. V. Kovalevskaia, did not lose its importance in the process. However, it recedes somewhat into the background, since it was found that it did not guarantee correctness in the solution of boundary value problems, that is, it did not guarantee the possibility of finding an approximate solution if the boundary conditions were 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 20t