Yurii Alekseevich Mitropolskii's name is often transliterated as Mitropolsky or Mytropolsky and occasionally as Mytropolskiy or Mitropolskiy. His father, Aleksei Savvich Mitropolskii, had attended St Petersburg University but was called up for military duty in 1914. When his son Yurii Alekseevich was born, Aleksei Savvich was serving at the front. After he was demobbed in 1919 he went with his family to Kiev where Yurii Alekseevich was brought up from the age of two. As a child he worked in a factory in Kiev. He entered the Faculty of Mechanics and Mathematics of Shevchenko Kiev State University in 1938 but on 22 June 1941 the German armies invaded their former allies pushing rapidly east into Soviet lands. At first their main advance was aimed towards Moscow, but by August they made a strong push in the south deep into the Ukraine heading towards Kiev. The University was evacuated from Kiev before the German troops reached the city and Mitropolskii was sent to the front. He was recalled from the front to continue his education at the Department of Physics and Mathematics at Kazakh University in Alma-Ata (renamed Almaty in 1991). Alma-Ata is the Soviet version of the Kazakh name Almaty for the capital of Kazakhstan, meaning "Father of Apples". It took this name in 1921 having previously been named Verny. Kazakh Al-Farabi State University was very new when Mitropolskii studied there, the University being founded in 1934.
Mitropolskii graduated from the Kazakh University in 1942 after studying there for six months. After graduating he attended the Ryazan Military Artillery School and then, from 1943 until the end of the war, he was sent to the front where he commanded an artillery intelligence platoon. The authors of [41] write for Mitropolskii's ninetieth birthday:-
For services in battle, he was awarded two Orders of the Red Star and military medals. Mitropolskii retains vivid memories of wartime, and he still remembers his comrades in arms and even details of the army life quite well.
He continued his military service until he was demobbed in 1946 when he began research at the Institute of Constructive Mechanics of the Academy of Sciences of the Ukraine, working under Nikolai Nikolaevich Bogolyubov. He was awarded his Candidate's Degree (equivalent to a Ph.D.) in 1948 for his dissertation on the problem of resonance phenomena in non-linear oscillatory systems with slowly varying parameters. His approach to the problem used the Krylov-Bogolyubov asymptotic methods. He continued to work for his doctorate (equivalent to the habilitation) and he was awarded this in 1951 for his thesis Slow processes in non-linear oscillatory systems with many degrees of freedom. In this impressive piece of work he studied problems of non-linear mechanics and mathematical physics which involved investigating non-stationary phenomena in non-linear oscillatory systems. He moved to the Institute of Mathematics of the Academy of Sciences of the Ukraine in 1951 and, two years later, he was appointed head of the Department of Mathematical Physics and Non-linear Oscillation Theory.
From 1951 Mitropolskii taught in the Faculty of Mechanics and Mathematics at Kiev University, where he was named as professor in 1954, and continued teaching there when made Director of the Institute of Mathematics in 1958. He held the post of Director of the Institute for 30 years, expanding the work of the Institute. Volodymyr Petryshyn writes in [36]:-
During Yu Mitropolskii's directorship (1958-88), the Institute experienced a great expansion in research personnel and mathematical disciplines, and an improvement in the quality of research.
Anatoly Samoilenko was a student of Mitropolskii's who obtained his Ph.D. in 1963. Following this, he worked with Mitropolskii on many joint mathematical projects and, when Mitropolskii retired from the directorship of the Institute in 1988, Samoilenko took over the directorship.
In [37] Volodymyr Petryshyn summarises Mitropolskii's work as follows:-
Mitropolskii has made major contributions to the theory of oscillations and nonlinear mechanics as well as the qualitative theory of differential equations. He further developed asymptotic methods and applied them to the solution of practical problems. He extended the Krylov-Bogolyubov symbolic method to nonlinear systems and extended asymptotic methods in the theory of nonlinear mechanics. Using a method of successive substitutes, he constructed a general solution for a system of nonlinear equations and studied its behaviour in the neighbourhood of the quasi-periodic solution. He also successfully applied the averaging method to the study of oscillating systems with slowly varying parameters.
The authors of [9] list seven main areas in which Mitropolskii made significant contributions:-
In 1955 Mitropolskii and Bogolyubov published a monograph on asymptotic methods in nonlinear oscillations. In particular this book contained the results the authors had obtained during the ten years from 1945 to 1955. Solomon Lefschetz begins a review with the following paragraph:-
The present book is the fourth or fifth major treatise published in recent years by Soviet scientists on the general topic of non-linear oscillations, which serves to indicate the great value which is attached in the USSR to this general topic. The general program of the book is not too far from the program of the 1937 Krylov-Bogolyubov monograph [Introduction to non-linear mechanics (1937)]. However, although the book is addressed primarily to physicists and engineers, its mathematical treatment is most careful, which was by no means the case with the 1937 monograph. The book is also much more orderly and most readable: an excellent contribution in every respect.
This work was to lead to further advances by the Kiev school, in particular they applied asymptotic methods to partial and functional differential equations. An English translation of the second Russian edition of the book (containing an additional chapter on single-frequency oscillations in systems with many degrees of freedom) appeared as Asymptotic methods in the theory of non-linear oscillations in 1961. A French translation appeared in 1962 with a German translation three years later. The method developed by the authors and presented in this and later editions of the monograph have come to be known as the KBM method (Krylov-Bogolyubov-Mitropolskii). This book was the first of many books written by Mitropolskii, the majority co-authored with his former doctoral students. The authors of [8] list 31 monographs published by Mitropolskii between 1955 and 2005. Among the single authored texts we mention: Nonstationary processes in non-linear oscillatory systems (1955); Problems in the asymptotic theory of non-stationary oscillations (1964); Lectures on the method of averaging in non-linear mechanics (1966); The method of averaging in nonlinear mechanics (1971); Nonlinear mechanics. Asymptotic methods (1995); Non-linear mechanics. Monofrequency oscillation (1997); and Methods of non-linear mechanics. A first textbook (2005). A reviewer of the 1964 monograph writes:-
The book is written for readers interested in the application of the techniques described. Asymptotic solutions of differential equations are worked out in great detail, the author always being willing to go the second mile with the reader in obtaining the inherently complicated formulas that arise. A large number of physical problems are presented, again in careful and lengthy detail.
Let us also quote from the Preface of the 1971 monograph:-
We deal with the method of averaging in nonlinear mechanics. We include numerous results of further development and generalization of the basic ideas of N N Bogolyubov. We give various algorithms, schemes and rules for constructing approximate solutions of equations with small and large parameters, and obtain examples which in many cases graphically illustrate the effectiveness of the method of averaging and the breadth of its application to various problems which are, at first glance, very disparate. The theorems that we include reveal the depth and mathematical rigour of the method of averaging. We discuss the basic trends and developments of the method of averaging, and as illustrations we give typical examples of nonlinear oscillatory systems, revealing the effectiveness of the method.
Among the many co-authored works we mention Lectures on the application of asymptotic methods to the solution of partial differential equations (1968) co-authored with his former student Boris Illich Moseenkov, Lectures on the methods of integral manifolds (1968) co-authored with his former student Olga Borisovna Lykova, Lectures on the theory of oscillation of systems with lag (1969) co-authored with his former student Dmitrii Ivanovich Martynyuk, Asymptotic solutions of partial differential equations (1976) co-authored with his former student Boris Illich Moseenkov, Periodic and quasiperiodic oscillations of systems with lag (1979) also co-authored with D I Martynyuk, Mathematical justification of asymptotic methods of nonlinear mechanics (1983) co-authored with his former student Grigorii Petrovich Khoma, Group-theoretic approach in asymptotic methods of nonlinear mechanics (1988) co-authored with his former student Aleksey Konstantinovich Lopatin, and Asymptotic methods for investigating quasiwave equations of hyperbolic type (1991) co-authored with his former students G P Khoma and Miron Ivanovich Gromyak. We give three examples of complimentary comments from reviewers of these texts:-
This list of works will already have given the impression that Mitropolskii had many outstanding research students. In fact the full list of his Ph.D. students is quite remarkable containing almost 100 names. He attracted many international students to the Institute and the list of his Ph.D. students contains students from Vietnam, Uzbekistan, Georgia, Bulgaria, and Yugoslavia. As remarkable is the fact that 500 students at the Institute of Mathematics of the Academy of Sciences of the Ukraine obtained a Ph.D. during the years that Mitropolskii was the director. Certainly he worked to have schools covering a wide range of topics in the Institute such as algebra, the theory of random processes, function theory, functional analysis, and mathematical physics.
We should also mention his important contribution to mathematics as an editor of several different journals. Some of these were Ukrainian journals such as the differential equations journal Differentsial'nye Uravneniya, while others were international journals such as the International Journal of Nonlinear Sciences and Numerical Simulations, the journal Nonlinear Analysis, the journal Nonlinear Dynamics, and the International Journal of Nonlinear Mechanics. He was also interested in the history of mathematics and served on the editorial board of the History of the Ukrainian Academy of Sciences and Essays on the development of mathematics in the USSR. To illustrate this interest let us mention his book The Institute of Mathematics: Academy of Sciences of the Ukrainian SSR written jointly with V V Strok and published in 1988.
Mitropolskii was elected to the Academy of Sciences of the Ukraine in 1961, to the Bologna Academy of Sciences (1971), and to the Academy of Sciences of the USSR in 1984. He was also honoured by the award of the A M Lyapunov Gold Medal in 1987. He was a speaker at the International Congress for Mathematicians in Edinburgh in 1958, in Stockholm in 1962, in Moscow in 1966, in Nice in 1970, in Vancouver in 1974, in Warsaw in 1983, in Berkeley in 1986, and in Kyoto in 1990. In 1965 he was awarded the Lenin Prize:-
... for his outstanding achievements in the theory of nonlinear differential equations and nonlinear oscillations.
He was also honoured with the award of the Krylov Prize (1969), the Bogolyubov Prize (1994), State Prizes of the Ukraine (1980 and 1996), and the Lyapunov Gold Medal (1986):-
... for the development of asymptotic methods in nonlinear mechanics.
For the same work he was awarded the Silver Medal of the Czech Academy of Sciences (1978):-
... for services to science and mankind.
He was made a Hero of Ukraine in January 2007 and, on the occasion of his ninetieth birthday, he was presented with the V I Vernadskii Gold Medal by the President of the National Ukrainian Academy of Sciences.
As to his personal characteristics, his colleagues write of his:-
... extraordinary creative energy, vigour, and optimism.
References for Yurii Alekseevich Mitropolskii
Books:
A N Bogoljubov (ed.), Yurii Alekseevich Mitropolskii: a bibliography : Compiled by M N Kreknina (Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 1977).
V K Kraineva and Ya A Matviishin, Yurii Alekseevich Mitropolskii : With an introduction by A N Bogolyubov, O B Lykova and A M Samoilenko, Biobibliography of Scientists of the Ukrainian SSR 'Naukova Dumka' (Kiev, 1987).
Articles:
60th birthday of the Ukrainian SSR Academy of Science's Academician Yu A Mitropolskii (Russian), Godishnik Vissh. Uchebn. Zaved. Tekhn. Mekh. 11 (1) (1976), 7-9.
Academician Jurii Olekseeovich Mitropolskii on the occasion of his sixtieth birthday (Bulgarian), Teoret. i Priloz. Meh.8 (1) (1977), 9-10.
Academician Yu A Mitropolskii (on the occasion of his seventieth birthday) (Russian), Ukrain. Mat. Zh. 39 (1) (1987), 3-4.
Academician Yu A Mitropolskii - hero of socialist labour (Russian), Vestnik Akad. Nauk SSSR (5) (1987), 131-132.
Academician Yu O Mitropolskii (on the occasion of his eightieth birthday) (Ukrainian), Ukrain. Mat. Zh. 49 (1) (1997), 3-4.
V I Arnold, V S Vladimirov, VV Kozlov, E F Mishchenko, Yu S Osipov, B E Paton, A N Sisakyan, A D Sukhanov, L D Faddeev, and K V Frolov, Yurii Alekseevich Mitropolskii (on the occasion of his ninetieth birthday) (Russian),Uspekhi Mat. Nauk 62 (4)(376) (2007), 179-185.
V I Arnold, V S Vladimirov, VV Kozlov, E F Mishchenko, Yu S Osipov, B E Paton, A N Sisakyan, A D Sukhanov, L D Faddeev, and K V Frolov, Yurii Alekseevich Mitropolskii (on the occasion of his ninetieth birthday), Russian Math. Surveys 62 (4) (2007), 829-835.
D Bainov, Jurii Alekseevich Mitropolskii, on the occasion of his 60th birthday (Bulgarian), Fiz.-Mat. Spis. B'lgar. Akad. Nauk. 20 (53) (2) (1977), 174-175.
N N Bogoljubov and V S Koroljuk, Ju O Mitropolskii's studies in the field of nonlinear oscillation theory (Russian), Ukrain. Mat. Z. 29 (1) (1977), 3-14, 141.
N N Bogoljubov, V S Koroljuk and A M Samoilenko, Jurii Alekseevich Mitropolskii (on the occasion of his sixtieth birthday) (Russian), Uspekhi Mat. Nauk 32 (1)(193) (1977), 217-228.
N N Bogolyubov, E F Mishchenko and A M Samoilenko, Yurii Alekseevich Mitropolskii (on the occasion of his seventieth birthday) (Russian), Uspekhi Mat. Nauk 42 4(256) (1987), 193-195.
M O Bogolyubov and O B Likova, Yurii Oleksiiovich Mitropolskii (on the occasion of his eightieth birthday) (Ukrainian), in The Institute of Mathematics. Outlines of its development (Ukrainian), (Natsional. Akad. Nauk Ukraini, Inst. Mat., Kiev, 1997), 147-155.
S Djadkov, Yu A Mitropolsky is sixty, Acta Tech. CSAV 21 (6) (1976), 621-622.
K V Frolov, E F Mishchenko, O A Oleinik, Yu S Osipov, A, M Samoilenko and V S Vladimirov, Yurii Alekseevich Mitropol'skii (on his eightieth birthday) (Russian), Uspekhi Mat. Nauk 52 (1) (1997), 237-239.
K V Frolov, E F Mishchenko, O A Oleinik, Yu S Osipov, A, M Samoilenko and V S Vladimirov, Yurii Alekseevich Mitropol'skii (on his eightieth birthday), Russian Math. Surveys 52 (1) (1997), 237-239.
N P Erugin, V S Koroljuk and O B Lykova, Jurii Alekseevich Mitropolskii (on the occasion of his 60th birthday) (Russian), Differencial'nye Uravnenija 13 (1) (1977), 177-184.
N P Erugin, V S Koroljuk and O B Lykova, Yurii Alekseevich Mitropolskii (on the occasion of his seventieth birthday) (Russian), Differentsial'nye Uravneniya 23 (1) (1987), 3-9.
N P Erugin, Ju D Sokolov, S F Fescenko and O B Lykova, Jurii Alekseevich Mitropolskii (Russian), Differencial'nye Uravnenija 3 (1967), 158-166.
I V Gaishun, V A Il'in, N A Izobov, V S Korolyuk, V N Koshlyakov, V F Kravchenko, I A Lukovskii, A A Martynyuk, V M Millionshchikov, E F Mishchenko, S I Pokhozhaev, N Kh Rozov, A M Samoilenko, A N Sharkovskii, and A B Vasilev, Yurii Alekseevich Mitropolskii (Russian), Differentsial'nye Uravneniya 44 (12) (2008), 1711-1713.
I V Gaishun, V A Il'in, N A Izobov, V S Korolyuk, V N Koshlyakov, V F Kravchenko, I A Lukovskii, A A Martynyuk, V M Millionshchikov, E F Mishchenko, S I Pokhozhaev, N Kh Rozov, A M Samoilenko, A N Sharkovskii, and A B Vasilev, Yurii Alekseevich Mitropolskii, Differential Equations 44 (12) (2008), 1776-1778.
I V Gaishun, N A Izobov, V A Il'in, V S Korolyuk, V N Koshlyakov, I A Lukovskii, V M Millionshchikov, E F Mishchenko, S I Pokhozhaev, N Kh Rozov, A M Samoilenko, A N Sharkovskii, and A B Vasileva, Yurii Alekseevich Mitropolskii (A Tribute in Honor of His Ninetieth Birthday) (Russian), Differentsial'nye Uravneniya 43 (1) (2007), 1-9.
I V Gaishun, N A Izobov, V A Il'in, V S Korolyuk, V N Koshlyakov, I A Lukovskii, V M Millionshchikov, E F Mishchenko, S I Pokhozhaev, N Kh Rozov, A M Samoilenko, A N Sharkovskii, and A B Vasileva, Yurii Alekseevich Mitropolskii (A Tribute in Honor of His Ninetieth Birthday), Differential Equations 43 (1) (2007), 3-10.
V M Gluskov, O S Parasjuk, V S Koroljuk and O B Lykova, Jurii Alekseevich Mitropolskii (Russian), Ukrain. Mat. Z. 19 (1) (1967), 3-8.
L Hatvani, Yurii Alekseevich Mitropolskii, founder of ICNO, in Proceedings of the Eleventh International Conference on Nonlinear Oscillations (Budapest, 1987), 14-17.
V A Il'in, N A Izobov, A A Martynyuk and A M Samoilenko, Yurii Alekseevich Mitropolskii (on the occasion of his eightieth birthday) (Russian), Differ. Uravn. 33 (1) (1997), 3-5.
V A Il'in, N A Izobov, A A Martynyuk and A M Samoilenko, Yurii Alekseevich Mitropolskii (on the occasion of his eightieth birthday), Differential Equations 33 (1) (1997), 1-3.
Juri Alekseevich Mitropolskii (on the occasion of his sixtieth birthday) (Russian), Mat. Fiz. Vyp. 22 (1977), 3-4.
J Kurzweil, On the sixtieth birthday of Yu A Mitropolskii (Czech), Pokroky Mat. Fyz. Astronom. 22 (1) (1977), 40-41.
V Lakshmikantham, A A Martynyuk and J H Dshalalow, Personage in science : Academician Yu A Mitropolskii, Nonlinear Dyn. Syst. Theory 6 (4) (2006), 309-318.
O Limarchenko and J-H He, Personage in science : Academician Yury Mitropolsky, Int. J. Nonlinear Sci. Numer. Simul. 1 (1) (2000), 3-5.
Obituary for Yurii Alexeevich Mitropolskii (1917-2008), Nonlinear Dyn. Syst. Theory 8 (4) (2008), 407-410.
On the 80th Birthday of Academician Yu A Mitropolskii (Russian), Ukrainskii Matematychnii Zhurnal 49 (1) (1997), 3-4.
On the 80th Birthday of Academician Yu A Mitropolskii, Ukrainian Mathematical Journal 49 (1) (1997), 1-2.
V Petryshyn, Mathematics, Encyclopaedia of Ukraine (Toronto-Buffalo-London, 1993), 339.
V Petryshyn, Mytropolsky, Yurii, Encyclopaedia of Ukraine (Toronto-Buffalo-London, 1993).
A M Samoilenko et al., Yurii Oleksiiovich Mitropolskii (on the occasion of his ninetieth birthday) (Ukrainian), Neliniini Koliv. 10 (1) (2007), 4-5.
A M Samoilenko et al., Yurii Oleksiiovich Mitropolskii (on the occasion of his ninetieth birthday), Nonlinear Oscil. (N. Y.) 10 (1) (2007), 1-3.
A M Samoilenko, Yu M Berezanskyi, V S Korolyuk, V M Koshlyakov, I O Lukovskyi, O M Sharkovskyi, M L Horbachuk, V L Makarov, M O Perestyuk, Yu I Samoilenko, O O Stepanets, P M Tamrazov, Yu Yu Trokhymchuk and V V Sharko, On the ninetieth birthday of Yurii Alekseevich Mitropolskii (Russian), Ukrainskii Matematychnii Zhurnal 59 (2) (2007), 147-151.
A M Samoilenko, Yu M Berezanskyi, V S Korolyuk, V M Koshlyakov, I O Lukovskyi, O M Sharkovskyi, M L Horbachuk, V L Makarov, M O Perestyuk, Yu I Samoilenko, O O Stepanets, P M Tamrazov, Yu Yu Trokhymchuk and V V Sharko, On the ninetieth birthday of Yurii Alekseevich Mitropolskii, Ukrainian Mathematical Journal 59 (2) (2007), 153-157.
A M Samoilenko and V G Kolomiets, On Yu O Mitropolskii's contribution to the development of the asymptotic methods of nonlinear mechanics (Ukrainian), Ukrain. Mat. Zh. 49 (1) (1997), 5-10.
A M Samoilenko and O B Lykova, Development of the methods of nonlinear mechanics in the works of Yu A Mitropolskii (Russian), Ukrain. Mat. Zh. 39 (1) (1987), 5-13.
N Stojanov and D Bainov, Academician Jurii Alekseevich Mitropolskii (on the occasion of his 60th birthday) (Bulgarian), Godishnik Vissh. Uchebn. Zaved. Prilozhna Mat. 12 (1) (1976), 9-11.
The awarding of the A M Lyapunov Gold Medal to Yu A Mitropolskii (Russian), Vestnik Akad. Nauk SSSR (2) (1987), 136.
The studies of Yu A Mitropolskii on the field of the theory of nonlinear oscillations and the theory of nonlinear differential equations (Russian), in Problems of the asymptotic theory of nonlinear oscillations 275 'Naukova Dumka' (Kiev, 1977), 7-14.
A B Vasilev, I V Gaishun, N A Izobov et al., Yurii Alekseevich Mitropolskii (Russian), Differ. Uravn. 44 (12) (2008), 1711-1713.
A B Vasilev, I V Gaishun, N A Izobov et al., Yurii Alekseevich Mitropolskii, Differ. Equ. 44 (12) (2008), 1776-1778.
A B Vasileva et al., Yurii Alekseevich Mitropolskii (on the occasion of his ninetieth birthday) (Russian), Differ. Uravn. 43 (1) (2007), 3-10.
A B Vasileva et al., Yurii Alekseevich Mitropolskii (on the occasion of his ninetieth birthday), Differ. Equ. 43 (1) (2007), 1-9.
V S Vladimirov, E F Mishchenko, O A Oleinik, Yu S Osipov, A M Samoilenko and K V Frolov, Yurii Alekseevich Mitropolskii (on the occasion of his eightieth birthday) (Russian), Uspekhi Mat. Nauk 52 1(313) (1997), 237-239.
Yurii Alekseevich Mitropolskii (on the occasion of his 75th birthday) (Russian), Ukrain. Mat. Zh. 44 (1) (1992), 3-4.
Yurii Alekseevich Mitropolskii (on the occasion of his seventieth birthday) (Russian), Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. (1) (1987), 88-89.
Yurii Aleksiiovich Mitropolskii (on the occasion of his sixtieth birthday) (Ukrainian), Visnik Kiev. Univ. Ser. Mat. Mekh. No. 19 (1977), 143-144.
Yurii Oleksiiovich Mitropolskii (Ukrainian), Neliniini Koliv. 11 (3) (2008), 292.
Yurii Oleksiiovich Mitropolskii, Nonlinear Oscil. (N. Y.) 11 (3) (2008), 305-306.
Yurii Oleksiiovich Mitropolskii (Ukrainian), Ukrain. Mat. Zh. 60 (9) (2008), 1155-1156.
Yurii Oleksiiovich Mitropolskii, Ukrainian Math. J. 60 (8) (2008), 1347-1348.
Mark G Krein's father was involved in the wood trade which meant that the family had little money. The fact that the family were Jewish meant that Krein grew up in an atmosphere of persecution. This was also highly significant for Krein's subsequent career, for discrimination against Jews in the Ukraine was bad and, by misfortune for Krein, was particularly bad in Odessa where he lived from the age of 17.
Krein showed a remarkable talent for mathematics at a young age. By the time he reached 14 years of age he was already attending research seminars in mathematics in Kiev. However, Krein never completed his undergraduate degree for he left his home in Kiev when he was 17 years old and ran away to Odessa. Despite the lack of an undergraduate degree, his talents were clearly visible to the mathematicians at Odessa University and, in 1926, Krein was accepted for doctoral studies under Chebotaryov.
Chebotaryov himself was not in the happiest of positions for, after working at Odessa University, he had accepted a permanent post in Moscow in 1924. The political situation surrounding this appointment induced him to leave Moscow and return to Odessa after only a few months. Krein therefore began his doctoral studies well aware of the difficulties of the time.
In 1928 Chebotaryov left Odessa and became professor at Kazan University. Krein completed his doctorate at Odessa in the following year and remained on the staff at the university building up one of the most important centres for functional analysis research in the world. During this time he worked on topics such as Banach spaces, the moment problem, integral equations and matrices, and on spectral theory for linear operators. As Osilenker writes in [28] and [29]:-
In the creative legacy of M G Krein a large place is occupied by the moment problem and the study of associated Jacobian matrices....
Krein's work on the moment problem, which played such an important part in his mathematical development, is discussed in detail in [25] and [26].
Kolmogorov had laid the foundations for the study of extremal problems in 1935 and Krein began to work on extremal problems for the class of differentiable periodic functions. Together with N I Akhiezer, Krein made a major contribution to this field in 1937.
In 1941 Krein had to leave Odessa when the university was evacuated as the German armies advanced. He was appointed as professor of theoretical mechanics at Kuibyshev Industrial Institute but he returned to Odessa in 1944. However, soon after taking up his post again he was dismissed. Gohberg writes in [13]:-
[Krein] was accused of Jewish nationalism, presumably for having had too many Jewish students before the War. This accusation was certainly included in his classified file and was presumably held against him all his life. ... He was not allowed to have Jewish students and was deprived of a university base.
Not only was Krein dismissed but the whole of the functional analysis school at Odessa was closed down. Potapov, who had been one of Krein's non-Jewish students, tried hard to influence the university authorities to reverse their decisions. He wrote:-
The leaders of the university have to correct their mistakes and to revive immediately the famous traditions of the Odessa School of Mathematics. The Faculty of Physics and Mathematics has to become active again, as a really creative centre of scientific and mathematical thought in our city ....
Krein was not reinstated, however, but held the chair of theoretical mechanics at Odessa Marine Engineering Institute from 1944. Also from 1944, he held a part-time post as head of the functional analysis and algebra department at the Mathematical Institute of the Ukranian Academy of Science in Kiev. This latter position came to a sudden end in 1952 when he was dismissed for a second time. Officially the reason given was that he did not live in Kiev, but in reality it seems more likely that the accusation of Jewish nationalism in his classified file was again the reason.
Although the 1940s must have been difficult times for Krein, his mathematical research did not suffer. Among other important work, Krein wrote eight papers on harmonic analysis and representation theory in the 1940s. In these papers he studied the use of algebras of operators applied to obtain results on positive definite kernels and functions. He also studied analysis on homogeneous spaces and duality theorems.
From 1954 until his retirement Krein occupied the chair of theoretical mechanics at the Odessa Civil Engineering Institute. During the 1940s and 1950s there were many unsuccessful attempts to have Krein and his students reinstated at Odessa University but all attempts failed.
During the many years of very active research Krein had around him a group of very active mathematicians although this group was based more on an informal arrangement than that of a proper research group. This group provided him with strong support, frequently meeting in Krein's own house. For example one of the Jewish student he ahd supervised before World War II was Livsic and he, like Krein, was not welcome back at Odessa University after the War. Livsic, again like Krein, did return to Odessa, teaching at the Hydrometerological Institute until 1957 and forming part of Krein's unofficial research group in Odessa. As well as Krein's house, the group often met at the Scientists' Club in Odessa and, in [16], Thomas Kailath describes a visit he made to Odessa where he talked at the Scientists' Club at the end of 1984.
Despite the support of those around him, Krein did have to work at a mathematical disadvantage in one sense, however, for he was not allowed to travel abroad and so was unable to attend international conferences that many mathematicians feel to be almost essential for someone so preeminent at an international level.
International recognition came to Krein despite his inability to visit centres of mathematical research around the world. In 1968 the American Academy of Arts and Sciences elected him an honorary member, and he was elected a Foreign Member of the National Academy of Sciences in 1979. Krein received many other honours, but perhaps the most prestigious award made to him was the Wolf Prize in Mathematics in 1982. After the years of discrimination on account of being Jewish it must have been particularly pleasing to receive this Prize from Israel. The citation for the prize summarises well the contribution that Krein made to mathematics:-
His work is the culmination of the noble line of research begun by Chebyshev, Stieltjes, Sergei Bernstein and Markov and continued by F Riesz, Banach and Szego. Krein brought the full force of mathematical analysis to bear on problems of function theory, operator theory, probability and mathematical physics. His contributions led to important developments in the applications of mathematics to different fields ranging from theoretical mechanics to electrical engineering. His style in mathematics and his personal leadership and integrity have set standards of excellence.
Gohberg writes of the quality and style of Krein's work in [13]:-
He is the author of more than 270 papers and monographs of unsurpassed breadth and quality. ... A profound intrinsic unity and close interlacing of general abstract and geometrical ideas with concrete and analytical results and applications and characteristic of Krein's work.
Despite a life of persecution, in which he often feared arrest, Krein remained enthusiastic, friendly and kind, showing great mathematical generosity towards his students and colleagues. Towards the end of his life, however, the struggle which he continually had to make took its toll and he suffered from depression which became worse after the death of his wife.
Honours awarded to Mark Krein
1. Speaker at International Congress 1966
2. Wolf Prize 1982
3. St Petersburg Mathematical Society Honorary Member
List of References (31 books/articles)
Naum Il'ich Akhiezer's father was the medical doctor for the district in and around Cherikov. The town of Cherikov, called Cherykau in Belarusian, had around 5500 inhabitants when Naum Il'ich was born. His brother Alexander Il'ich Akhiezer was ten years younger than Naum Il'ich and went on to become a famous theoretical physicist. Naum Il'ich attended the local in Cherikov, graduating in 1918. Then he taught mathematics and physics at the Lemelsk school-commune for four years before beginning his university studies. Perhaps we should make clear that, following the Revolution, Ukraine did not have any universities, but it contained Institutes of Public Education which filled a similar role.
Akhiezer studied at the Kiev Institute of Public Education, entering in 1922 on a three-year course of study. However, he was an exceptional student and was able to complete the full course of study by December 1923 having taken around eighteen months. This is even more remarkable when one realises that he was also teaching at one of the local Kiev schools during the time he was studying at the Institute. In many ways he went far beyond what was expected of any student [6]:-
While preparing for the examinations he was also able, in this short time, to prepare resumes of a number of the basic courses; these were subsequently used by many students. His accelerated passage through the course did not prevent him from taking an active part in a number of scientific seminars. To this same period dates his first enthusiasm for the theory of , which he studied from the original memoirs of Jacobi in Latin. Subsequently he was to return several times to this theory, and to make masterly use of it in various problems of analysis. At the same period he began studying the celebrated doctoral dissertation of N E Zolotarev, and certain other work by classic native authors.
He met a fellow undergraduate Mark Grigorievich Krein who had entered the Kiev Institute of Public Education at the age of fourteen. Akhiezer and Krein became friends and around ten years later collaborated on mathematical projects. Akhiezer graduated from the Kiev Institute of Public Education in 1924. In addition to his school teaching, at this time Akhiezer was assisting Dmitry Aleksandrovich Grave by conducting practical classes for him at one of the Higher Educational Institutes in Kiev. Akhiezer became a research student of Grave's in 1925 and worked on his dissertation [6]:-
At that time Grave had begun to occupy himself with questions of mechanics and applied mathematics, and he led his pupils in the same direction. Akhiezer's research was on the theory of functions of a complex variable and its applications to aerodynamics. In March, 1928 he defended his dissertation 'Aerodynamical Investigations' (Ukrainian), which had appeared in the journal 'Trudy Piz.-Mat. Otdela Ukrain. Akad. Nauk.' We remark that it was in this paper that the formula was first given for mapping a doubly-connected domain, bounded by polygons, onto a circular annulus.
After undertaking research with Grave, in 1928 he joined the staff at Kiev University and the Kiev Aviation Institute and taught there until 1933. During this period he worked on the theory of functions publishing many papers in Russian, French and German. Examples of his papers from his time in Kiev are: On polynomials deviating least from zero (Russian) (1930), On the extremal properties of certain fractional functions (Russian) (1930), On a minimum problem in the theory of functions, and on the number of roots of an algebraic equation which lie inside the unit circle (Russian) (1931), and Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen (3 parts, 1932-1933). In the first of these papers Akhiezer solved the problem of finding the polynomial with three fixed coefficients deviating least from zero on a given interval. This extended results by Chebyshev who had solved the problem for polynomials with one fixed coefficient, and by Zolotarev for polynomials with two fixed coefficients.
From 1933 Akhiezer worked at Kharkov University. There he joined the Kharkov School of function theory and soon became its leading member. He was appointed to the Chair of the Theory of Functions in 1933 and, two years later, was appointed director of the Mathematical and Mechanical Research Institute after Sergei Bernstein moved to Leningrad. He served as President of the Kharkov Mathematical Society for more than 25 years and was elected to the Ukrainian Academy of Sciences in 1934. Akhiezer had never been awarded a doctorate since academic degrees of Master of Science and Doctor of Science had been abolished in the USSR in 1918. However, they were reintroduced in 1934 and, in 1936 Akhiezer was awarded an honorary degree of Doctor of Physical-Mathematical Sciences. In around 1935, Kolmogorov laid the foundation for a new study, namely that of the extremal problem for a class of functions. In 1937 Akhiezer, working with his friend M G Krein, solved the extremal problem for the class of differentiable periodic functions. The two mathematicians collaborated on writing the book On some problems of the theory of moments (Russian) published in 1938. Akhiezer continued to work on this topic and was later to solve the extremal problem for the class of analytic functions. He published The classical moment problem and some related topics in analysis in 1961 (Russian, English translation 1965). A review begins:-
The moment problem is of importance in several areas of mathematics, e.g., certain problems of analysis such as quadrature formulas, , orthogonal polynomials, the interpolation problem for functions of a complex variable, quasi-analytic classes of functions and monotone functions, spectral theory of operators, etc. ... The present book serves as a solid foundation for the application of the moment problem to the above fields ...
During World War II, as his contribution to war work, he moved to the Alma-Ata Mining and Metallurgy Institute in 1941 where he remained until 1943 when he transferred to the Moscow Power Institute. He returned to his Chair of the Theory of Functions at Kharkov University in 1947. Between 1941 and 1945 he did not publish any papers but from 1945 onwards he was able to publish again with papers such as On some inversion formulae for singular integrals (1945), The general theory of Chebyshev polynomials (1945), On some properties of entire transcendental functions of exponential type (1946) and the book Lectures on the Theory of Approximation in 1947. This important book was awarded the Chebyshev Prize in 1949. Antoni Zygmund writes:-
This is an interesting and valuable book representing various aspects of the theory of approximation of functions in the real domain. ... The presentation is clear and elegant.
Andrey Kolmogorov, in a review of this book, described Akhiezer's monographs as:-
... masterpieces of mathematical literature, combining width and clarity of general conception with virtuosity in the handling of details.
The book was translated into English and published in 1956, being reprinted in 1992.
His main work was on function theory and approximation theory, building on the results of Chebyshev, Zolotarev and Markov. Volodymyr Petryshyn comments that:-
His most outstanding work consisted of deep approximation results in the constructive function theory, including the solution of the problem of Zolotarev.
Boris Verkin, the founder and first director of the Institute for Low Temperature Physics and Engineering of the Academy of Sciences of Ukraine, was a specialist in experimental physics and held mathematics in high esteem. He persuaded several mathematicians including Akhiezer to join the Institute in the early 1960's and Akhiezer became head of the Department of the Theory of Functions at this Institute.
Akhiezer's later work, in addition to that on the theory of moments, included joint work with Sergei Bernstein on completeness of sets of polynomials. Akhiezer wrote 150 papers and 10 books, one of which was the important Theory of Operators in . This work, written jointly with Israel M Glazman, was first published in Russian in 1950. It was translated into English and published as two volumes, in 1961 and 1963. In 1966 the authors produced a major revision and augmentation of the text for a second Russian edition, and they produced a third edition in two volumes in 1977, 1978. This third Russian edition was translated into English and published in 1981. The two English volumes published in 1961 and 1963 were reprinted as a single volume in 1993.
He also contributed to the history of mathematics with an important book on Sergei Bernstein and his work. He also edited the collected works of Pafnuty Lvovich Chebyshev, Andrei Andreyevich Markov, Aleksandr Mikhailovich Lyapunov and Nikolay Yakovlevich Sonin.
Akhiezer was highly praised as a teacher [6]:-
He comes to his teaching with the greatest sense of responsibility, and thoroughly prepares his courses, usually in written form. He reads these lectures with great animation, infecting the audience with his unaffected enthusiasm and desire to reveal the beauty of mathematical constructions. His great personal charm, sparkling wit and cheerfulness, and his open and friendly attitude towards the young, have always attracted talented young people to Akhiezer, a great number of whom have developed into good mathematicians who are working in various cities of the USSR.
In [3] some of his other contributions are mentioned:-
Naum Il'ich had a wide range of interests, never ceased to organise mathematician teams in Kharkov Polytechnical Institute and Kharkov University. More than that, he took many far reaching initiatives to raise the new generation of mathematicians (setting up the well known 27-th mathematics school, the Kharkov State University based External youth mathematics school). His pedagogical gift and lecturer's skill were famed in Kharkov higher education institutions.
The authors of [5] write:-
Many generations of students at the University of Kharkov heard Akhiezer's remarkable lectures and were astonished by their elegance and originality. These lectures had a significant formative influence on many mathematicians. He was gladdened by the success of young mathematicians, who always found in him a supporter. He had a brilliant and fascinating personality; his intellectual interests were unusually wide, and his energy and enthusiasm did not forsake him until the very last day.
References