# Georgii Nikolaevich Polozii

### Quick Info

Born
23 April 1914
Transbaikal, Russia
Died
26 November 1968
Kyiv, Ukraine

Summary
Georgii Polozii was a Russian mathematician who mostly worked in pure mathematics such as complex analysis, approximation theory and numerical analysis.

### Biography

Georgii Nikolaevich Polozii was the son of Nikola Andriyovych Polozii and Olena Hryhorivna Sviatkina. We note that his name appears with many different spellings depending of the transliteration. For example, Polozii is often written Polozhii or Polozhiy. His patronymic name Nikolaevich appears in various spelling, but perhaps the most confusing is Mikolaevich meaning that often his name appears as G N Polozii or G M Polozii. In fact some of his mathematical papers have the author's name as G N Polozii, while others have G M Polozii.

Georgii Nikolaevich's paternal grandfather had been forced to move to the village of Kapustin Yar on the Astrakhan steppes after the Russo-Turkish war of 1877-1878. Georgii Nikolaevich's father, one of eighteen children, became a telegraph operator on the Trans-Siberian railway. At the time that his son Georgii Nikolaevich was born, he was working at the 37th junction of the Trans-Baikal railway, a part of the Trans-Siberian railway built between 1895 and 1905. This was in the Chita region, east of Lake Baikal, about 900 km from Irkutsk, the first large city one reaches travelling west along the Trans-Siberian railway from Chita. Nikola Andriyovych and Olena Hryhorivna had five children, four girls and one son, Georgii Nikolaevich, the subject of this biography. He was the youngest of the five children.

During his stay at the school he found a special inclination for physical and mathematical sciences.
Polozii would have wished to continue immediately to university studies but this was not allowed at that time before obtaining work experience. The largest employer in Nizhny Baskunchak was the Bassol plant for the extraction of salt from Lake Baskunchak and processing it. The company's products included table salt, iodized salt and industrial salt. He worked in management at the Bassol plant until January 1933, then in the following month he enrolled as a first-year student in the Faculty of Physics and Mathematics of Saratov State University which had been founded in 1919. He was taught by Georgy Petrovich Boev who had a huge influence on Polozii's development as a teacher, mathematician and scientist. To understand this influence let us look briefly at Boev's methods of teaching as described by B V Gnedenko, who was also taught by Boev at Saratov State University, in [5]:-
[Boev] based his approach on the premise that mathematics for engineers was not an end in itself but a means of studying engineering subjects and a method to be applied in future practical work. Therefore, mathematicians must teach what is necessary for the student and must present the material in such a way as to show him why he is studying it. But in order to understand which mathematical methods and corresponding solutions are required by specialists [in a given field], mathematicians themselves must study some of the field's basic features and its important problems.
While still an undergraduate, Polozii showed his ability to do scientific and mathematical analysis so, after he graduated in 1937 with the highest honours, he was appointed as an assistant at the Department of Mathematics of the Saratov Highway Institute. He began to undertake research but was not in a position to have advice from a well-qualified advisor. He wrote some notes based on his first research attempts and sent them to Mstislav Vsevolodovich Keldysh hoping that they might be worth publishing in the Doklady Akademii Nauk SSSR. They were rejected by M V Keldysh who wrote the following to Polozii on 12 December 1937 (see [12]):-
The results of your work can be obtained almost directly from Schwartz's lemma ... and so, because of the simplicity with which the results of the note can be obtained, it should not be published in 'Doklady Akademii Nauk'. However, the wording of your note is of interest in addition to the facts presented and, after simplification of the proofs, they could be published in the section of small notes ... . I sent your three notes on the theory of filtration to Pelageia Yakovlevna Kochina with a request to review them and give feedback. She is an expert in these matters. Upon receipt of her opinions, I will inform you about them. It would be nice if you came to Moscow on vacation in order to discuss the issues related to preparing your dissertation. Then I will give you the literature you are interested in.
These notes were never published, but the connection that was made between Polozii, M V Keldysh and P Ya Kochina proved highly significant and, eventually, M V Keldysh would become Polozii's advisor for his doctoral dissertation. Polozii was appointed as an assistant at the Department of Mathematical Analysis of Saratov State University, where he became actively engaged in research. His studies were interrupted, however, by the country becoming involved in major wars. In August 1939, Russia and Germany had signed a secret pact, the so-called Ribbentrop-Molotov pact, to divide Poland between them. Soviet troops invaded Poland on 17 September 1939 and by 29 September Poland was partitioned between Russia and Germany. The Ribbentrop-Molotov pact also gave Soviet forces the freedom to invade the Baltic States with German agreement not to oppose this, but the pact was not known at the time. In September-October 1939 the Soviets pressured the Baltic States to allow them to set up military bases there. On 30 November 1939 the Soviet Union invaded Finland resulting in what is now known as the Winter War. Despite superior military strength, the Soviets made little progress. On 1 February 1940, Polozii went to the front as a volunteer but, following the Moscow Peace Treaty in March 1940, fighting ended and a few months later, on 11 November 1940, Polozii returned to Saratov State University.

At this stage the Ribbentrop-Molotov pact meant life in the Soviet Union was relatively normal but this changed dramatically on 22 June 1941 when Germany broke the non-aggression pact and invaded the Soviet Union. Two days later, on 24 June 1941, Polozii again volunteered for the front to fight in what became known as the Great Patriotic War. He was in the infantry, had the rank of junior lieutenant and became commander of a rifle platoon. On 3 September 1941 he was wounded near the village of Mykhailivsky, but returned to the front after treatment. Much more serious, however, was the wounds he received on 22 February 1942 in the fighting against the invasion by the German army near Nelidovo, Nelidovsky District, Kalinin region. He spent two years being treated for his injuries, during which time he underwent seven operations. At the beginning of 1944 he was declared unfit for further military service and he returned to Nizhny Baskunchak.

Polozii, now disabled, was faced with difficult decisions. He had to decide whether to attempt to return to his mathematical career or to try to make his way in a different profession, perhaps one in which his disabilities would be less of a hinderance. He turned for advice to his former professor Georgy Petrovich Boev who strongly supported his return to an academic career. With Boev's support, on 6 April 1944 Polozii was appointed as an assistant in the Department of Mathematical Analysis of Saratov State University. He only had a first degree and to make a career as a professor he had to first obtain a Candidate's Degree, equivalent to a Ph.D., and then a doctorate, essentially equivalent to the habilitation. The first step was the Candidate's degree and he began working for that advised by G P Boev. In October 1944 he was promoted to a position equivalent to a senior lecturer or associate professor.

In 1946 Polozii defended his doctoral dissertation Integral Representations of Continuously Differentiable Functions of a Complex Variable (Russian) in which he had begun to develop a new approach to generalised analytic functions. In this dissertation he was the first to establish the Cauchy formula for (p, q)-analytic functions. Let us quote from [1] at this point:-
The following results, obtained by G N Polozii and which are the basis of the whole theory of $(p, q)$ analytic functions, indicate the close structural and qualitative connection of these functions with the classical theory of families of solutions of the Cauchy-Riemann system:

1. The concept of an integral with respect to the conjugate variable, and the introduction of a differentiation operation inverse to the operation of integration with respect to the conjugate variable.

2. The concept of conjugate kernels and the construction of a generalised Cauchy integral and a generalised Cauchy-type integral; the differential properties of $(p, q)$-analytic functions.

3. The classification of isolated singularities, and the fundamental theorem on residues.

4. The properties of uniqueness and $(p, q)$-continuation of $(p, q)$-analytic functions; isolation of the roots of the equation $f(z) = A = \text {const}$.

5. Theorems on conservation of the domain and congruence of the boundaries; theorem on the conservation of a Schlicht neighbourhood.
We have listed papers by Polozii at THIS LINK.

We have listed books by Polozii at THIS LINK.

We note that the five results quoted above appear in the papers 1., 3., and 12. in our list of Polozii's papers. They are all discussed in Polozii's book The Method of Summary Representation for Numerical Solution of Problems of Mathematical Physics (Pergamon Press, 1965). This book is 5. on our list of Polozii's books where we give its preface, introduction and extracts from some reviews.

In 1949 Polozii was appointed to the Taras Shevchenko National University of Kyiv as an associate professor of mathematical analysis. This university, founded in 1834 as the Saint Vladimir Royal University, had been the Mykhailo Drahomanov University from 1920 to 1932, before being renamed the Taras Shevchenko University of Kyiv in 1939. After one year, he was appointed associate professor of mathematical physics, holding this position for the year 1950-51. He became Head of the Department of Mathematical Physics in October 1951. The previous Head of the Department of Mathematical Physics at Kyiv had been Nikolai Nikolaevich Bogolyubov but he left to take up the position of Head of the Department of Theoretical Physics, a new department, in the Steklov Mathematical Institute in Moscow. From November 1952 to February 1953 Polozii served as dean of the Faculty of Mechanics and Mathematics at the University of Kyiv.

Although Polozii was Head of the Department of Mathematical Physics, he was not a professor; that would only be possible after he had obtained a doctorate. On 25 June 1953 he defended his doctoral dissertation On some methods of function theory in continuum mechanics which he defended at the Steklov Mathematical Institute of the Academy of Sciences of the USSR. His opponents were Mikhail Alekseevich Lavrentev and Pelageia Yakovlevna Polubarinova Kochina. Lavrentev wrote in his report on the dissertation:-
The author gives a solution to the classical theory of mixed elasticity, when normal displacements and tangential stresses are put on the boundary. The problem is solved for a polygonal region. This problem attracted the attention of many who work on elasticity, but the complete solution has only been obtained in the work of G N Polozii.
Kochina wrote in her report that:-
... there is a group of problems associated with the mathematical theorem of G N Polozii, on the motion of boundary points. This theorem itself is elegant and is of mathematical interest, and there is already further work developing from it. And on the other hand, it is very useful in filtration theory, because it allows one to give estimates of some quantities of immediate interest.
After the award of his doctorate, Polozii received the title 'Professor of Mathematical Physics' in 1954. From that time, he began to work to ensure the creation of a Computing Centre at Kyiv University, which he successfully acomplished. During this period, the USSR began to pay considerable attention to the development of computational mathematics and in many higher education institutions began to create departments of computational mathematics. In December 1957, the Department of Computational Mathematics was established in the Faculty of Mechanics and Mathematics of the University of Kyiv. The composition of the department was formed of the best young teachers of other departments in the Faculty of Mechanics and Mathematics and talented scientists from other scientific institutions. The rector of the University appointed Pavel Stepanovich Bondarenko as temporary head of the new Department while a competition was set up to select a permanent head. Polozii was chosen as the first permanent head of the new Department taking up the post in 1958. He continued in this position until his death in 1968.

We have already seen Polozii's major pure mathematical contributions but let us summarise his contribution: the theory of functions of a complex variable, approximation theory, and numerical analysis. He also made major contributions to mathematical physics and applied mathematics, in particular working on the theory of elasticity. Between 1962 and 1966 Polozii developed the theory for a new class of $(p, q)$ analytic functions. Petryshyn, writing in [7], summarises Polozii's work:-
Original results in the theory of functions of a complex variable were obtained in the 1950s and 1960s by G Polozii of Kyiv, who introduced a new notion of $p$-analytic functions, defined the notion of derivative and integral for these functions, developed their calculus, obtained a generalised Cauchy formula, and devised a new approximation method for solution of problems in elasticity and filtration. His results were further developed by his students ...
In approximation theory Polozii worked mainly with the aim of developing effective methods to solve boundary value problems which arise in mathematical physics. His work here produced the method of summary representation. Let us quote from [6] where the work of Polozii and his Department in this area is put into a broad context. One of the main areas of research in the Department of Computational Mathematics was:-
... approximate methods for solving problems of mathematical physics. Almost all modern mathematical methods used to solve the most interesting of scientific and technical problems, both from a theoretical and, especially, from a practical point of view, are methods of finding an approximate solution of the problem. These methods are divided into two groups - analytical and numerical. The first makes it possible to obtain a solution in the form of a formula that allows for its qualitative analysis. But, unfortunately, analytical methods are developed only for a relatively narrow range of fairly simple problems that very seldom reproduce real processes. More accurate and, consequently, more complex mathematical models are implemented by numerical methods. The latter give a solution only at certain points in the scope of the problem, and this is the main disadvantage of numerical methods. In addition, classical numerical methods work only in limited areas of problem definition. But these shortcomings could not cover the significant advantages of numerical methods in comparison with analytical: applicability to an incomparably wider range of tasks and the relative ease of implementation, especially with the help of computers. Apparently, every qualified mathematician who has worked and is working with approximate methods has been thinking about how to combine the positive properties of analytical and numerical methods. For the boundary value problems of the equations of mathematical physics there is a model which established these processes; this was created by G M Polozii and his students and followers. The method of summary images developed by him (or, as it is also called, the method of $P$-transformations) made it possible to construct solutions of the corresponding finite-difference problems in a closed form and to write them in the form of formulas of summary images. Depending on the type of boundary conditions, these formulas are either explicit or determined by a small number of unknown parameters. These parameters can be calculated by solving the corresponding system of linear algebraic equations. Another feature of the method should be noted: it is ideal for implementation on a computer with parallel processing. In this respect, the fate of the method is reminiscent of the fate of the fast Fourier transform, which was discovered in 1925, but was most widely used only with the widespread use of computers. Both of these methods were ahead of their time. Georgii Nikolaevich himself developed only the theoretical foundations of the method of summary images, and his students were engaged in the practical implementation of these developments and its further improvement.


Among the many honours given to Polozii, let us mention his election as a corresponding member of the Ukrainian Academy of Sciences in 1967 and several USSR medals and awards, for example in 1959 and 1964.

Over the last years of his life, Polozii worked to create a Faculty of Computational Mathematics and Cybernetics. Sadly he died at the age of 54 only a year before this Faculty was established. He was buried in Kyiv at Baykovo Cemetery.

The conference 'Differential Equations, Computational Mathematics, Function Theory and Mathematical Methods of Mechanics' was held at the Faculty of Mechanics and Mathematics of Kyiv University on 23-24 April 2014 to celebrate the centenary of Polozii's birth.

### References (show)

1. P I Chalenko and V S Chemeris, Mathematical research at Kyiv State University, Ukrainian Mathematical Journal 19 (1967), 706-714.
2. Dedicated to the 100th anniversary of the birth of Georgiy Polozhiy, KN News Ukraine (22 June 2014).
http://kunews.knu.ua/100-richchyu-vid-dnya-narodzhennya-georgiya-polozhiya-prisvyachena/
3. L Fox, Review: The Method of Summary Representation for Numerical Solution of Problems of Mathematical Physics, by G N Polozhii, J. London Math. Soc. 42 (1) (1967), 374-375.
4. Georgii Nikolaevich Polozii (Russian), Vycisl. Prikl. Mat. (Kyiv) No. 9 (1969).
5. B V Gnedenko and Z Khalil, The Mathematical Education of Engineers, Educational Studies in Mathematics 10 (1) (1979), 71-83.
6. History of the Department, Department of Computational Mathematics, Faculty of Cybernetics, National Taras Shevchenko University of Kyiv.
[No longer available on the Web.]
7. Mathematics, Encyclopedia of Ukraine (Toronto-Buffalo-London, 1993).
8. V V Ivanov and I I Lyashko, Georgii Nikolaevich Polozhii, Ukrainian Mathematical Journal 20 (1968), 742-743.
9. V V Ivanov and I I Lyashko, Georgii Nikolaevich Polozhii (Ukrainian), Ukrainskii Matematicheskii Zhurnal 20 (6) (1968), 856-857.
10. Polozhiy, Georgii Nykolayovych, Department of Computational Mathematics, Faculty of Cybernetics, National Taras Shevchenko University of Kyiv.
[No longer available on the Web.]
11. G N Polozhii, The Method of Summary Representation for Numerical Solution of Problems of Mathematical Physics (Pergamon Press, 1965).
12. A M Samoilenko et al, Georgii Nikolaevich Polozhii (Ukrainian), Mathematics Mechanics 2 (32) (2014), 56-58.
13. A Solomon, Review: The Method of Summary Representation for Numerical Solution of Problems of Mathematical Physics, by G N Polozhii, Mathematics of Computation 21 (97) (1967), 123-124.