PEOPLE OF SOVIET SCIENCE

ON THE 60TH BIRTHDAY OF BORIS PAVLOVICH DEMIDOVICH

On March 2, 1966, it was 60 years since the birth and 30 years of scientific and pedagogical activity of the well-known Soviet mathematician, Professor of Moscow University Boris Pavlovich Demidovich.

B. P. Demidovich was born in Belorussia, in the town of Novogrudok of the former Minsk Province. His father, Pavel Petrovich Demidovich, was a teacher at the Novogrudok town school and, in addition to his pedagogical work, devoted much attention to collecting materials of Belorussian folklore. Subsequently, for his works on Belorussian ethnography, he was elected a corresponding member of the ethnography section of the Society of Lovers of Natural Science, Anthropology, and Ethnography at Moscow University. His mother, Olimpiada Platonovna Demidovich (née Plyshevskaya), before marriage was a rural schoolteacher and later raised the children.

After finishing secondary school, B. P. Demidovich in 1923 entered the physics and mathematics section of the pedagogical faculty of Belorussian State University (Minsk). Belorussian University had been organized only in 1920, and during this period instruction in the physics and mathematics section was conducted by persons who had come mainly from secondary schools. At BSU a great influence on Boris Pavlovich was exerted by Docent V. K. Dydyrko, who at that time taught courses in analytic geometry, higher algebra, and the theory of differential equations, as well as by Professor A. A. Mikhailovsky, who had come from Leningrad and taught courses in theoretical mechanics and general astronomy. As a student, Boris Pavlovich, together with M. P. Dorofeenko, organized the publication of lecture notes by Docent I. S. Pyatosin on mathematical analysis, by Docent V. K. Dydyrko on the integration of partial differential equations, and others.

After graduating from BSU (1927), Boris Pavlovich worked for four years as a teacher of mathematics in secondary educational institutions.

In 1931 B. P. Demidovich was admitted first to the preparatory, and then to the regular postgraduate program of the Research Institute of Mathematics at Moscow University, where he worked under the supervision of V. V. Nemytskii and took an active part in the seminar of V. V. Stepanov and V. V. Nemytskii on qualitative methods in the theory of differential equations. His candidate dissertation, “On the existence of an integral invariant on a system of periodic orbits,”

closely connected with the subject matter of this seminar and was a substantial achievement in solving the difficult problem of conditions for the existence of an invariant measure in a dynamical system.

In 1936 Boris Pavlovich Demidovich received the academic degree of Candidate of Physical and Mathematical Sciences and was invited to Moscow University, where he works at the present time, while at the same time teaching at other higher educational institutions in Moscow.

The scientific activity of Boris Pavlovich is connected chiefly with questions of the theory of stability in the sense of Lyapunov and the asymptotic behavior of solutions of ordinary differential equations. On the basis of the totality of his published works, B. P. Demidovich was awarded the academic degree of Doctor of Physical and Mathematical Sciences in 1963, and in 1965 he was confirmed by the Higher Attestation Commission in the academic rank of professor.

In the works of B. P. Demidovich [11, 12] an important critical case for applications is investigated: stability in the sense of Lyapunov of the linear system

\[ \frac{d x}{d t} = [A + B(t)]x, \tag{1} \]

where \(x\) is an \(n \times 1\) vector; \(A\) and \(B(t)\) are \(n \times n\) matrices; \(B(t) \to 0\) as \(t \to \infty\), under the assumption that the characteristic roots \(\lambda_j(A)\) of the matrix \(A\) satisfy the condition

\[ \lambda_1(A) = 0,\quad \operatorname{Re}\lambda_j(A) < 0 \quad (j \ne 1). \]

In the extensive work [22] the existence and properties of solutions \(\tilde{x}=\tilde{x}(t,\mu)\), bounded on \((-\infty,+\infty)\), of the nonlinear system

\[ \dot{x} = A x + \varphi(t,x,\mu), \]

are studied, where \(A\) is a constant matrix; \(\varphi(t,x,\mu)\) is the nonlinear term and \(\mu\) is a scalar parameter. Here, on the basis of the application of Wazewski’s topological principle, the well-known theorem of Boole is generalized.

In the article [23] sufficient conditions are given for the existence of a limiting regime of the nonlinear system

\[ \frac{d x}{d t} = F(x) + G(t), \]

where \(x, F(x), G(t)\) are \(n \times 1\) vectors, with \(G(t)\) being \(T\)-periodic.

In the works [29, 30] a system of the form

\[ \frac{d y}{d t} = P(t)y + f(t,y;\mu,\omega) \tag{2} \]

is considered, with a variable \(n \times n\) matrix \(P(t)\), where \(f\) is the nonlinear term; \(\mu\) is a small parameter and \(\omega\) a large parameter. Under the assumption that the solutions of the homogeneous system

\[ \frac{d x}{d t} = P(t)x \]

possess the dichotomy property, sufficient conditions were obtained for the existence of a solution \(\hat{y}=\hat{y}(t,\mu,\omega)\) of system (2), bounded on \((-\infty,+\infty)\) and depending continuously on \(\mu\) and \(\omega\). The stability of this solution was investigated. Moreover, if the matrices \(P\) and \(f\) are almost periodic in \(t\), then it is shown that the solution \(\hat{y}\) is also almost periodic in \(t\). These results generalize the theorem of Farnell, Langenhop, and Levinson on the existence and stability of forced harmonic oscilla-

nonlinear system of the form (2) for the case \(P(t)=\operatorname{const}\) (J. of Math. and Phys., 29, No. 1, 300—302, 1950).

A series of works [31—33, 35] is devoted to dissipative systems

\[ \frac{d\mathbf{x}}{dt}=\mathbf{f}(\mathbf{x},t) \tag{3} \]

\[ \bigl(\mathbf{f}(\mathbf{x},t)\in C_{t\mathbf{x}}^{(0,1)}(I_t^+\times E_{\mathbf{x}}^n)\bigr), \]

all of whose solutions \(\mathbf{x}(t)\) are uniformly ultimately bounded,

\[ \varlimsup_{t\to\infty}\|\mathbf{x}(t)\|<R \]

(\(R\) is a constant). Using the characteristic numbers \(\lambda_j(\mathbf{x},t)\) \((j=1,\ldots,n)\) of the generalized symmetrized Jacobi matrix

\[ \widetilde{J}_s(\mathbf{x},t)=\frac{1}{2}\{Af'_{\mathbf{x}}(\mathbf{x},t)+[Af'_{\mathbf{x}}(\mathbf{x},t)]^{*}\} \]

(\(A\) is a constant positive definite \(n\times n\)-matrix), sufficient conditions are established for the dissipativity of system (3), and also, in the case of almost periodicity in \(t\) of its right-hand side \(\mathbf{f}(\mathbf{x},t)\), sufficient conditions for the existence of a unique almost periodic solution of this system.

The article [38] contains a generalization of Lyapunov’s stability criterion for regular systems in the case of nonpositivity of the characteristic exponents of its linear part.

B. P. Demidovich is the author of a number of manuals [10, 20, 28, 34, 36] (partly jointly with others) for courses in higher mathematics, differential equations, numerical mathematics, quantum mechanics, etc. In particular, he is the author of the widely known university problem book on mathematical analysis [14], which has gone through a number of editions and has been translated abroad.

Since 1935 Boris Pavlovich has been a member of the Moscow Mathematical Society and has been a regular participant in mathematical congresses (beginning with the Third) and conferences, where he has presented original reports.

The scientific, pedagogical, and public activity of B. P. Demidovich has been recognized with government awards: the Order of the “Badge of Honor” and medals.

We wish Boris Pavlovich Demidovich health and further creative successes.

N. P. ERUGIN, V. V. NEMYTSKII

LIST OF PRINTED WORKS OF B. P. DEMIDOVICH

1936

1. On the existence of an integral invariant on a set of periodic points. DAN SSSR, vol. II (XI), No. 1 (87), 11—13.

1938

1. On certain sufficient conditions for the existence of an integral invariant. Matem. sb., 3 (45), No. 2, 291—310.

1939

1. Articles in BSE, ed. 1, 43, and 46.

1944

1. A Collection of Problems in Higher Mathematics. Publishing House of the MVTU; co-editor and member of the authors’ collective.

1945

1. On a theorem from a course in analysis. Collection of Methodological Works of the Department of Higher Mathematics of the MVTU. Publishing House of the MVTU, 47—52 (in manuscript form).
2. On the integration of total differentials. Ibid., 81—84.

1948

1. Periodic solutions of a nonlinear system of second-order ordinary differential equations with right-hand sides periodic with respect to the independent variable. DAN SSSR, 61, No. 4, 601—603.

1949

1. The existence of periodic solutions for a certain nonlinear system of ordinary differential equations. Vestnik MGU, No. 2, 13—25.
2. Oscillations of a rod bent along an arc of a circle. Engineering Collection, vol. V, issue 2, 112—132.
3. A Short Course in Higher Mathematics (jointly with V. A. Kudryavtsev). GTTI, ed. 1, 406 (GTTI, ed. 2, 1959, 432; Fizmatgiz, ed. 3, 1962, 528). Translated into Chinese.

1950

1. On one critical case of stability in the sense of Lyapunov. DAN SSSR, 72, No. 6, 1005—1008.

1951

1. On stability in the sense of Lyapunov of a linear system of ordinary differential equations. Matem. sb., 28 (70), No. 3, 659—684.

1952

1. Differential equations. BSE, 2nd ed., 14, 520—526 (jointly with V. V. Nemytskii and A. N. Kolmogorov).
2. A Collection of Problems and Exercises in Mathematical Analysis (for universities and physics-and-mathematics pedagogical institutes). GTTI, ed. 1, 516 (GTTI, ed. 2, 1954, 511; GTTI, ed. 3, 1956, 511; Fizmatgiz, ed. 4, 1959, 511; Fizmatgiz, ed. 5, 1962, 544). Translated into Romanian and Chinese.
3. On certain properties of the characteristic exponents of a system of ordinary linear differential equations with periodic coefficients. Uch. zap. MGU, issue 163, Mathematics, vol. IV, 123—132.

1953

1. On a case of almost periodicity of a solution of an ordinary differential equation of the first order. UMN, vol. VIII, issue 6 (58), 103—106.

1954

1. On a generalization of N. N. Bogolyubov’s averaging principle. DAN SSSR, 96, No. 4, 693—694.
2. On certain averaging theorems for ordinary differential equations. Matem. sb., 35 (77), issue 1, 73—92.
3. A simple proof of the mean-value theorem for harmonic functions. UMN, vol. IX, issue 3 (61), 213—214. Translated into Romanian.

1955

1. Differential Equations (lecture notes). Publishing House of the Military Engineering Academy named after F. E. Dzerzhinsky, 200.

1956

1. Bounded solutions of a certain nonlinear system of differential equations. Proceedings of the 3rd All-Union Mathematical Congress, vol. 52, Moscow.

1. On bounded solutions of a certain nonlinear system of ordinary differential equations. Matem. sb., 40 (82), No. 1, 73–94.

2. On the existence of a limiting regime of a certain nonlinear system of ordinary differential equations. Uch. zap. MGU, issue 181, Mathematics, vol. VIII, 3–12. Translated into English.

1957

1. On the boundedness of monotone solutions of a system of linear differential equations. UMN, vol. XII, issue 2 (74), 143–146.

1958

1. Methods of computational mathematics, part 1 (jointly with I. A. Maron). Publishing House of the Military Engineering Academy named after F. E. Dzerzhinsky (as a manuscript), 308.

1959

1. Methods of computational mathematics, part 2 (jointly with I. A. Maron and E. Z. Shuvalova), 300.

2. Problems and exercises in mathematical analysis for higher educational institutions (editor and member of the authors’ collective). Fizmatgiz, ed. 1, 472 (ed. 2, 1961; ed. 3, 1962; “Nauka,” ed. 4, 1966). Translated into English.

1960

1. Fundamentals of computational mathematics (jointly with I. A. Maron). Fizmatgiz, ed. 1, 660 (ed. 2, 1963; “Nauka,” ed. 3, 1966). Translated into Polish. Warszawa, 1965, Państwowe Wydawnictwo Naukowe (in press).

1961

1. On bounded solutions of a certain quasilinear system. DAN SSSR, 138, No. 6, 1273–1275.

2. Forced oscillations of a quasilinear system in the presence of a rapidly varying external force. PMM, vol. XXV, No. 4, 705–715.

3. On the dissipativity as a whole of a certain nonlinear system of differential equations. UMN, vol. XVI, issue 3 (99), 216.

4. On the dissipativity of a certain nonlinear system of differential equations. I. Vestnik MGU, No. 6, 19–27.

1962

1. On the dissipativity of a certain nonlinear system of differential equations. II. Vestnik MGU, No. 1, 3–8.

2. Numerical methods of analysis (jointly with I. A. Maron and E. Z. Shuvalova). Fizmatgiz, ed. 1, 368 (ed. 2, 1963, 400).

1963

1. On certain nonlinear systems with the Levinson D-property. Proceedings of the Symposium on Nonlinear Oscillations. Kiev, 156–160.

2. Mathematical foundations of quantum mechanics (lecture notes). Publishing House of the Military Engineering Academy named after F. E. Dzerzhinsky, 200.

1964

1. Summary of the report “Stability of solutions of a nonlinear system with a completely regular linear part,” Publ. Inst. für Angew. Math. und Mech., Berlin, 1964, 10–11.

1965

1. On one generalization of Lyapunov’s stability criterion for regular systems. Matem. sb., 66 (108), No. 3, 344–353.