# Alexander Semenovich Kronrod

### Born: 22 October 1921 in Moscow, Russia

Died: 6 October 1986 in Moscow, Russia

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**Alexander Kronrod**developed a great interest in mathematics from an early age and joined the student group organized by David Oskarovich Shklyarskii (1918-1942), who was an excellent teacher. Shklyarskii's general method was to lead students to find solutions to difficult problems on their own. Alexander won one of the prizes at the fourth Moscow Mathematical Olympiad in 1938 while still a member of Shklyarskii's group.

Let us note at this point that our biography is based largely on translations of various Russian biographies of Alexander Kronrod made by University of St Andrews student Yulia Yashneva.

Kronrod entered the Faculty of Mechanics and Mathematics of Moscow State University in 1938 and became an outgoing student, whose abilities were immediately recognised by a large number of academic staff. His teachers [2]:-

During his first year at the university Kronrod managed to solve an open question and publish his first independent work. Aleksandr Osipovich Gelfond, head of Mathematical Analysis Department, suggested a problem to his student group of which Kronrod was a member, regarding the possible structure of the set of points of discontinuity of a function which is differentiable at each point where it is continuous. Alexander Kronrod published his first article about this problem in 1939 in the journal I... were enthralled by his outstanding talent, enormous energy, range of activity, and his sometimes deliberately paradoxical statements - even by his appearance - he was tall and had a beautiful sonorous voice.

*zvestiya Akademii Nauk*.

World War II broke out on 1 September 1939 when German troops entered Poland. On 22 September, Russian troops entered Poland occupying Białystok and five days later Warsaw fell to the Germans and, following the Molotov-Ribbentrop non-aggression pact between Germany and the Soviet Union, Poland was partitioned between these two powers. The Molotov-Ribbentrop non-aggression pact meant that the initial years of the World War II had little effect on life in Moscow and, in particular, none on Kronrod for, despite his attempts to enlist for military service, he was rejected since graduate level students were exempted. Things changed dramatically on 22 June 1941 when Germany broke the non-aggression pact and invaded the Soviet Union. Graduate level students, including Kronrod were sent to a military academy and helped with building trenches around Moscow during first days of war against Germany. Nonetheless, he still wanted to enlist in the army and, after returning from the military academy, he reapplied and was accepted. He was injured twice and in 1943 he became unfit for duty [2]:-

He was awarded several medals, including the Order of Red Star and the Order of the Patriotic War.His military career was not easy. During the winter offensive of the Soviet army near Moscow, his bravery resulted not only in his receiving his first military decoration, but also his first severe injury. After he was wounded a second time in1943, his return to the army became out of the question. He preserved his ability to study mathematics, but not to fight. The last injury made him an invalid; its effects were felt throughout his life.

While being hospitalized after the wounds he suffered in 1943, he worked on a problem which had been suggested to him by M A Kreines before he began his military service. An answer was required to the following question:

Kronrod did not just provide an answer to the question but also categorised such permutations. He called permutations 'left' if they changed some convergent series into a divergent one. He called permutations 'right' if they changed some divergent series into a convergent one. He called permutations which were both 'right' and 'left', 'two-sided' permutations, and he called permutations which were neither 'right' nor 'left', 'neutral' permutations. A neutral permutation could not change the sum of a series. The last part of his work included criteria for belonging to the classes and application of the results to series with complex entries. His work was published in the journalSuppose a permutation on the set of natural numbers changes the sum of an infinite series into different sum. Is it possible that the same permutation changes a conditionally convergent series into a divergent one?

*Matematicheskii Sbornik*in 1945 and led to him receiving the first of two Moscow Mathematical Society awards for young scientists.

Kronrod still needed to complete the final year of his university course and resumed his studies in 1944 in the Faculty of Mechanics and Mathematics of Moscow State University. Nikolai Nikolaevich Luzin returned to teach at Moscow State University in February 1945. He announced he would give the lecture course 'Theory of function of two real variables' and also organise a seminar related to the topics of course. At this time Luzin was a legendary figure with an exceptional reputation as a teacher so Kronrod was proud to attend the course and the seminar. Kronrod and Adelson-Velskii became Luzin's final two students, and Kronrod, having no other advisors unlike Adelson-Velskii, was always proud of his status as "Luzin's last student'. Luzin gave Kronrod a signed copy of the French edition of his famous book

*The integral and trigonometric series*which Kronrod loved to show to every one.

Luzin suggested a problem for both Kronrod and Adelson-Velskii to work on. Could they prove by using methods of the theory of function of a real variable (without using integration methods) that each function

*f*(

*x*+

*iy*) =

*u*(

*x*,

*y*) +

*iv*(

*x*,

*y*) where

*u*(

*x*,

*y*) and

*v*(

*x*,

*y*) satisfy the Cauchy-Riemann conditions, can be expressed as a power series. Kronrod and Adelson-Velsky looked at this problem in a broader way and considered the following equations

∂

with positive functions *u*/∂*x*=*A*(*x*,*y*) ∂*v*/∂*y*, ∂*u*/∂*y*= -*B*(*x*,*y*) ∂*v*/∂*x**A*(

*x*,

*y*) and

*B*(

*x*,

*y*) and discovered the connection between smoothness of roots and of the coefficients of

*A*and

*B*(which are identically equal to 1 in the case of Cauchy-Riemann equations). Studying the behaviour of level curves of the functions of two variables

*u*(

*x*,

*y*) and

*v*(

*x*,

*y*) and establishing their maximum principle was one of the significant parts of their work. This work raised their interest in studying level curves of any continuous function of two variables and led to further results which they published later, for which they were both given awards by Moscow Mathematical Society.

Kronrod went further and explored linear variation and monotonicity of functions of two or more variables. Even though he considered only continuous functions, his results could be also applied to discontinuous functions. He also outlined the direction for further research which later was conducted by his students. His research in this area provided him with the foundation for his dissertation which he submitted for a Candidate's Degree (equivalent to a Ph.D.) in 1949. Examined by a committee consisting of Mstislav Vsevolodovich Keldysh, Andrei Nikolaevich Kolmogorov, and Dmitrii Evgenevich Menshov, his outstanding thesis was awarded a doctorate (equivalent in standard to a D.Sc. or habilitation) in Physics and Mathematics.

Development of the theory of functions of two variables, for which he used 'Kronrod's tree', can be seen as one of his most significant achievements. The importance of this idea can easily be seen since Vladimir Igorevich Arnold used Kronrod's tree to solve Hilbert's 13th problem in his Candidate's thesis in 1961. The next major problem which Kronrod tried to solve was the following:

Let

This had been an open question for many years and, despite several mathematicians attacking it, it had remained open. Kronrod, however, showed that the question had a positive answer, but he did not publish the solution. His reason was a sad one for pure mathematics, since he had decided that he would not undertake further research in the topic. It was a decision which he adhered to for the rest of his life.
*S*be a given surface with bounded Lebesgue area, parametrically embedded in**R**^{3}. Is it true that*S*has an asymptotic tangent plane almost everywhere (in the sense of the measure generated on*S*by Lebesgue area)?Even when studying at Moscow State University Kronrod has worked in the computing department of the Kurchatov Atomic Energy Institute. His reason for undertaking this work was purely financial since by this time he was married and had a wife and young son, born in 1943, to support. He was always someone who gave 100% to everything he did and at this time he devised an algorithm for solving linear second-order two-point boundary-value problems or tridiagonal linear systems which arise in the finite-difference solution of them. Although he found computational mathematics interesting, he was not ready to leave 'traditional' pure mathematics at that time. A few years on, however, things changed.

Kronrod was involved in building the first Russian computer 'Relay Computer RVM-1' with the engineer Nikolal Ivanovic Bessonov. We note that a significant fact about Kronrod's personality was his modesty since he said that Bessonov was the sole inventor. L D Landau and I V Kurchatov recommended Kronrod to A I Alikhanov, the Head of a new Moscow atomic institute (later named the Institute for Theoretical of Experimental Physics), and in 1949 Kronrod became Head of the Mathematical Department there. From that time on he fully devoted himself to computational mathematics. Unlike many mathematicians and scientists, Kronrod loved being an organiser so this position suited him. In addition he felt that computational mathematics was more useful and again this was a factor in his change of topic.

Apart from being involved in research work, Kronrod also organised his own student group in 1946-1953. Being an excellent teacher, he managed to create an enthusiastic and creative atmosphere within the group. Functions of one real variable, set theory and topology were some of the topics discussed at the meetings. He became a teacher for some outstanding mathematicians such as E V Glivenko, A A Milyutin, E M Landis (an author of [1], [2] and [3]), and R A Minlos.

In 1950-1955 Kronrod was mostly working on numerical solutions for physical problems and collaborating with his close friends Isaak Ya Pomeranchuk and Lev Landau. Kronrod was awarded the Stalin Prize and an Order of the Red Banner for his work at the Institute. Only in 1955 did he gain access to the computer M-2 (which was more efficient than the previous models) constructed by I S Bruk, M A Kartsev and N Ya Matyukhin in the laboratory at the Krzhizhanovskii Energy Institute.

Kronrod made a significant contribution to Artificial Intelligence, a new area in computing at that time. He believed that work should be focused around machine learning rather than solely numerical problems. He said [7]:-

Kronrod created a team of mathematicians and physicists to develop Artificial Intelligence. Firstly, they decided to create a programme for the difficult Russian card game called 'Durak'. This game was not selected at random. It does not have a thoroughly developed theory and has a simple position description. The code for this game worked only when there were spare cards left. When spare cards were used up, the computer was not powerful enough to complete the game because the 'tree' of possible solutions and strategies was too extensive to process. The team decided to leave the idea of creating a program to play this game and decided instead to write a chess playing program since it is also known worldwide. The program was developed by Adelson-Velsky, Arlazarov, Bitman and Uskov. Whenever the team struggled with recursion issues, Kronrod always helped to address them. They organized a meeting with the American team from Stanford University which created the best chess programme under the leadership of J McCarthy. The program of the Kronrod's team won with a score of 3:1. Kronrod is believed to be one of the first scientists to work in the area of Artificial Intelligence.Of course, I am convinced that a computer can think. That is, more precisely, that there is no such logical, mathematical, theoretical, etc. etc. tasks that future machines will not be able to do given time.

There was strict control in Kronrod's department. Mathematicians created solutions to physical problems on blank sheets, which were later sent to the coding team. An interesting fact was that there were more women than men in the coding team. Kronrod claimed that from his experience women worked better and provided more accurate results. Recruiting only of the best workers and paying higher wages and bonuses for error-free codes (bonuses accounted for 20% of wages) led to significant achievements made by the team. Bessonov dealt with the technical part and ensured the best performance of the hardware. In 1963, under the leadership of Kronrod, Bessonov rewrote the command system and doubled the power of the machine. Kronrod's department resembled an industrial factory with strict discipline and regular achievement of excellent results. Even though Soviet computers were less powerful than Western ones, Kronrod's team still managed to excel due to his programming skills.

Towards the end of the 1950s Kronrod became interested in economics, especially in issues regarding pricing. He noticed that the pricing principles in the Soviet Union were inherently inaccurate and influential Soviet economists agreed with him. A commission to consider pricing issues was created in the Ministry of Finance, which Kronrod was involved in during 1961-1962. He applied Leontief's matrices for input-output analysis for the Soviet economy. This quantitative part of the project was done with the RVM-1 computer and later with the M-20. After a while, Kronrod's student V D Belkin led the project.

In the 1960s Kronrod discovered a new area of interest for him: differential diagnostics for some illnesses with the use of computers. At the Gertsen Research Institute for cancer research P E Kunin, who was a physicist and a student of Kronrod, set up a laboratory for differential diagnostics for lung cancer and central pneumonia. When working there Kronrod achieved some significant results. Kunin's unexpected death, however, terminated the project. During that time Kronrod also organized mathematical classes in middle schools and developed teaching methods.

He, along with many other mathematicians, signed a letter of support for the mathematician and logician Alexander Esenin-Volpin, a son of the poet Sergei Esenin, who had been sent to an insane asylum for his anti-Communist views. This led to Kronrod receiving a reprimand from the Communist Party, and he was dismissed from the Moscow Institute for Theoretical and Experimental Physics in 1968. The reprimand was really used as an excuse for his dismissal, for the physicists thought that Kronrod was taking up precious computer time "playing games." Later that year he became the head of the mathematical laboratory at the Central Scientific Research Institute of Patent Information. He believed that there should be profound patent reforms in order to promote creativity and innovation in business. After failing to find common ground with the newly appointed director, Kronrod left the Institute.

The last place of work for him was at the Central Geophysical Expedition. He headed the laboratory which was processing data from drilling exploratory wells. He introduced new computing and numerical methods but this job was not a good match for his talents.

Kronrod was very sociable and had friends from completely different circles. He created optimal methods for developing oil and gas projects for his friend Lapuk, who was an influential person in oil and gas industry. He discussed arts with the famous actor Evstigneev and the script writer Nusinov. Being friends with famous doctors and surgeons, he was impressed by their work. He helped Bogdanov, a creator of anabol for cancer patients, to promote the medicine and make it easier access to it since it was imported from Bulgaria. However, anabol was expensive. Kronrod organized the production of similar medicine milil to help cancer patients. Later he was given an opportunity to do animal testing for his medicine in the laboratory at the Vishnevsky Institute. Kronrod, however, never used animal testing but instead carried out all the experiments on himself. Despite the fact that he had a deep knowledge of medicine, he did not have a medical degree. He never treated patients himself and sometimes even paid from his own money for their treatment by a professional medic. Nonetheless, a criminal case was brought against him and his card index file was taken from him due to the fact that milil was not approved by pharmacologists. The attorney on the case was diagnosed with cancer and his treatment required milil. Therefore, the case was dismissed and Kronrod's card index file returned. Soon after that Kronrod suffered a stroke. He could not read or write for some time. He finished working at Central Geophysical Institute. He became interested only in medicine and finished his career in mathematics. He suffered two strokes later and died on 6 October in 1986.

Alexander Kronrod was a very talented scientist and a good teacher. His student Peregudov said about him:-

I was so charmed by Alexander Semenovich that I could not really communicate with other people: I always compared people to Kronrod but so few people could reach his level.

**Article by:** *J J O'Connor* and *E F Robertson*

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