# Leonid Vitalyevich Kantorovich

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### Mathematician, economist and cyberneticist Academician of the USSR Academy of Sciences Nobel Prize winner in 1975

Alexander Nitusov

Leonid Vitalyevich Kantorovich[1]

Outstanding mathematician and economist, academician L.V. Kantorovich made a great contribution to the world science with a number of his fundamental achievements, among them:

- original theory of semi-ordered spaces in functional analysis, named K-spaces after Kantorovich;
- creation of a new domain in mathematics and economy focused on solution of optimisation tasks – it has been named “
*Linear programming*”; - methods of “Large-block” programming or organisation of computing processes and their influence on development of the architecture of computer systems.

With some of his ideas (“linear programming”, etc.) L.V. Kantorovich was definitely far ahead of his time, what was eventually recognised by the world’s scientific society. Thus Kantorovich became a joint winner of the 1975 Nobel Prize for economics, for his work on the optimal allocation of scarce resources (based on his applied mathematical research in economy).

Later in his career Kantorovich also was interested in computer engineering. Design of some experimental machines was based of his suggestions. His original concepts of the *large-block organisation of computing processes* noticeably influenced development of computer architecture. Recognition of his scientific contribution is in its turn a convincing proof of the strong influence, which Soviet mathematical school exerted upon the progress of computer science and engineering.

## Family and University

Leonid Vitalyevich Kantorovich was born on the 19th of January 1912 in St. Petersburg (Leningrad). His father Vitaliy Moiseevich Kantorovich was a popular doctor–venereologist and his mother, Paulina (Poline) Grigoryevna Zaks, played an important role in Leonid's upbringing.

There is no clear evidence that he was a child prodigy in his early years, however in 1926, when he entered mathematical faculty of the Leningrad University he was just 14. In 1930 he graduated (at the age of 18) and stayed at the university as a post-graduate. Kantorovich made his research under supervision of professor G.M. Fichtengolts. Famous mathematicians B.N. Delone [2], and V.I. Smirnov also were his teachers. In 1932 he was appointed as a lecturer at the university. His young age and appearance caused some funny episodes – the students first did not believe that the “youngster” was their lecturer and not a fellow-student. In 1934 he received a professorship and in 1935 was awarded a doctorate (“Doctor of physical and mathematical sciences”) without submission a thesis (he was exactly 23).

From 1930 till 1939 he also worked at the Leningrad Institute of Engineers of Industrial Construction, in parallel with his university activities.

## Scientific life

Leonid Kantorovich was undoubtedly a brilliant “born scientist”. Funny enough he was the only scholar in the family – his elder brother had chosen medical career (same as the father) and, in 1938, Leonid himself married Natalya Ilyina, who was also a doctor.

His systematic researches in mathematics Kantorovich started during the second university year. First he demonstrated interest to *theory of sets* and *theory of substantial functions*. Series of his works on *projective sets* and on *analytic operations with sets* was published in 1929-1930 in *Doklady AN SSSR*. It suggested, among others, solution of some problems of the theory of sets, which had been set up by academician N.N. Luzin. His report on those researches was also welcomed by the First All-Union Mathematical Congress, what confirmed high scientific level of his work.

Continuing his researches Kantorovich suggested new methods of *conformal mapping* and *variational method* for approximate replacement of partial derivative equations with systems of ordinary differential equations. The both methods were highly applicable in mechanics and engineering. They were presented in complete form in a monograph, “Approximate Solution Methods of Equations in Partial Derivatives”, published in joint authorship with V.I. Krylov in 1936. Later it was translated into some foreign languages. Both this book and its improved version, “Approximate methods of higher analysis” (1941) became classical works for computational mathematics.

At the same time he conducted theoretical research on *functional analysis* and initiated systematic study of *function spaces – partially ordered spaces*, which were subsequently named K-spaces (after Kantorovich). The results were published in *Doklady AN SSSR *1935-1936*.* In a scientific “free seminar” set up by V.I. Smirnov he also took leading position on *functional analysis* issues. That theory was also developed in Holland, Japan and the USA at approximately the same time. However, the first -collective- monograph, “Functional Analysis in Semi-ordered Spaces”, jointly composed by L.V. Kantorovich, Z. Vulich and A.G. Pinsker, was published only in 1950, when it was already outside of the focus of his scientific interests.

Kantorovich claimed that reasonable generalisation of a problem (or enlargement of some its fragments) could often be more efficient in its solution than thorough study of details. That was the way he solved some complex problems of the *theory of functions* set up by Moscow school of academician N.N. Luzin. He possessed a lucky ability to see -invisible- connections between, often remote, mathematical lows or phenomena, what always helped him to find advanced solutions.

However, mathematic school of Leningrad not only performed advanced theoretical studies but also constantly implemented their results in solution of purely applied problems. One could probably claim that one of Kantorovich’s most important discoveries came to being due to that tradition.

In 1938, the plywood production trust of Leningrad invited him, as a scientific expert, to one of its plants to inspect organisation of its manufacturing process and suggest possible ways of its improvement. Considering that the plywood, as building material, was popular in pre-war aircraft industry, importance of the problem would be scarcely doubted. Kantorovich found out that there were 5 different veneer-cutting lathes, cutting 8 types of veneer. Each lathe had its own performance rate for each kind of veneer. The core of mathematic problem (as it was described by V. Zalgaller[3]) consisted in finding optimum task distribution (between the lathes) for reaching the maximum of total production rate. Seeing that there was no available solution he soon suggested a successful method of his own.

Speaking in more rigorous mathematical terms Kantorovich noticed that *maximising linear form of multiple variables under numerous limitations, in form of linear equations and inequations,* was the problem he faced. He modified *method of Lagrange multipliers* and obtained a new one, characterised with usage of auxiliary coefficients named by him *“resolving multipliers”*. He implemented it in his solution scheme and subsequently realised that a great amount of various economic problems could be reduced to such schemes. Even more – the problems of that kind always accompanied need of *economic utilisation of sparse resources*.

Kantorovich shrewdly noticed that *nature* of those multipliers was purely *costs based*, what was another outstanding conclusion. Therefore, expanding the scheme onto macro-economy would create highly rational structure of economic indexation, what in its turn could essentially improve price-forming system of, excessively centralised, planned economy, and thus significantly increase its flexibility.

In 1939, having been inspirited by that impressive discovery, he published a monograph, “Mathematical Methods in Organising and Planning Production” describing a spectrum of economic problems that might be solved with new methods. In reality that was the founding of new mathematic domain later named *Linear Programming*. The publication itself became a historical document containing details of the discovery.

Publishing of the book had been supported by the university administration. It appeared already in 1939. However, not many copies were printed, as it was purely theoretical academic study, and they were mainly distributed among ministerial offices and organisations. Although the work did not encounter any directly negative reaction, no official response arrived from the economic officials at all. Of course, bureaucratic organisations normally deal with directives, instructions but not with mathematical theories, no matter how applied. Another possible reason consisted in “excessively advanced” character of the theory itself. The 1930-s were just the beginning formation phase of the Soviet national economy (with its rigorously centralised planning). As it was in the state of intensive progress and its structure was still “rather simple” its managing resources were focused more on consolidation and acceleration than flexibility. Also dominating perception of the economy as a subject of exclusively ‘ideology based’ direct descending planning, having rather dogmatic character that time, could cause suspicious attitude to such ‘mathematical intrusion’.

In 1940-1941 Kantorovich prepared another work, about wood-sawing industry, and one more on optimisation in transportation, jointly with M.K. Gavurin. Those works remained unpublished and, with the previously distributed one, unnoticed until 1949. That year he became a laureate of the “Stalin Premium”, awarded for his outstanding contribution to mathematics, in particular for his fundamental work, “Functional Analysis and Applied Mathematics”, published in 1948. Only then they were finally printed and his theory reached broad scientific (and other) audience. However, by that time Dantzig (Dantzig, George), Ford and Fulkerson of the USA had already published their own work solving the problem of *linear programming*.

In 1941 the Institute of industrial construction, where he was a professor, was transformed into High Technical School (HTS) of military engineering. With outbreak of the war he himself was assigned military rank of a major and evacuated with the institute from Leningrad to town of Yaroslavl (300 km North from Moscow). During the war he resided there[4] lecturing at the HTS and simultaneously doing various applied works for needs of defence. He also managed to compose a new course-book on *probability theory* oriented on military issues. It was published already in 1946.

In 1942 he set up a new principal field of perspective research, for which his big manuscript, “Economic calculations providing most rational usage of resources” became the cornerstone. Unfortunately its “way to recognition” was not simple and, same as his “linear programming”, it had to remain in shadow for a long time. In 1942 academician S.L. Sobolev, a person of influence in official mathematical circles, supported its presentation at the “Gosplan” (governmental head office on planning in national economy). However, it failed an approval both there and later at some scientific seminars of higher level.

That was obviously -and unfortunately- a wrong time for innovations. All national efforts were mobilised on the counter-advance of the Soviet Army and maximally quick restoration of industry, heavily destructed by the Nazi invasion of the 1941-1942. The state administration was primarily interested in stable rhythmic functioning of the economy.

Nevertheless, thirty years after, his name obtained world fame of the Nobel Prize winner for the research on *optimal allocation of scarce resources**.*

In 1942 Kantorovich published an article in *Doklady AN SSSR* (an academic journal) presenting, in abstract form, a method for solving transporting problems. The method implemented his “resolving multipliers” (as potentials). The problem itself was formulated as *transportation of masses in compact metric space*. Planning of railway transportation and levelling of an airfield surface were analysed in the article as examples. That article published both in Russian and English languages, was most probably the first source providing information on linear programming for foreign specialists. Anyway, much later famous American scientist G.B. Dantzig mentioned in his book “*Linear Programming and Extensions”*(Princeton University Press, Princeton (1963)) that, already in 1939 Kantorowich described almost everything on linear programming, what was known in America by 1960.

Kantorowich never interrupted his engagement in theoretical science, however, constant wartime concentration on practical issues, above all defence problems, noticeably hampered his progress in pure mathematics. Soon after the war he complained with obvious lagging behind the contemporary level and expressed doubts about possibility of regaining his advanced position in that domain. “Anyway, I wished to do something useful”, he commented, “So I felt that working out new methods in economical science was also important and promising field”[5][2].

In the post war years he continued active work on various *mathematical models of economy,* incorporating them in solution of large-scale problems of planning. He implemented them in (mathematical) foundations of price-formation methods and used for estimation of investments efficiency. He also dealt with non-linear programming. Although his “Mathematical revolution” (or counter-revolution?) was sharply rejected he continued researches (annoying “bureaucrats”). However, in 1945, he clearly felt approaching danger and had to temporarily freeze the work. “I even fell into a state of depression for a while and was not sure that ever will be able to return to economics” [2].

However, in 1958, Kantorovich was elected to the AS USSR, as corresponding member of the department of economics, philosophy and sciences of law. In 1964 he became a real member of the Academy, at the mathematical department.

The work on mathematical models in economy was presented again in 1959. That time response was much more favourable. General situation in the country essentially changed. He analysed problems of the choice of optimal parameters in technological processes, studied industrial transportation problems, efficiency of investments, forming of wholesale prices and general structure of economic criteria to be a basis of efficient economic calculations. It received broad, first Soviet and then world, recognition.

In 1965 he was awarded “Lenin Premium” (jointly with economists V.S. Nemchinov and V.V. Novozhilov).

By the beginning of the Sixties his scientific relations extended and obtained some new quality, he established contacts with foreign colleagues. His book on linear programming, written in 1939, was finally translated into English language and published abroad on initiative of professor Tjalling C. Koopmans (1910-1985) – a Dutch scientist who was working in the USA, after having been forced to move there by the Nazi invasion of Holland on the 10th of May 1940. Soon after, another his book (of 1959) also appeared abroad. Koopmans cooperated with Kantorovich for considerable time. He visited Soviet Union twice, in 1965 and in 1970, always accompanied and assisted by his wife Truu.

In 1975 Kantorovich and Koopmans shared the Nobel Prize on economy.

## Computers and programming

Lecturing in Yaroslavl Kantorowich simultaneously was doing some work in Moscow. During one of the visits he received an invitation from a mathematician Prof. Lyusternik (Lusternik Lasar' Aronovich), later one of the Soviet pioneers of programming, to a seminar on computing machinery. In reality the seminar had little chances to be very exciting. Most of the time they studied slow analytical machines and inefficient old-fashioned tabulators, which were able of -averagely- two additions per second and one multiplication within 5-8 seconds. Very little information was available on the first electronic computers, like American MARC-1 or MARC-II. Nevertheless he was very interested and began planning possible applications for computing facilities.

In 1944 Nazis’ siege of Leningrad, which lasted 900 days, was broken and Kantorovich returned there to head the Department of Approximate Calculations at the Leningrad Mathematical Institute (AS USSR). He also resumed work at the Leningrad University; a memorial plate has been placed on its building reading that L.V. Kantorovich worked there in 1930-1948.

In the middle of 1948 Kantorovich and the team of his collaborators were engaged as mathematicians in the state program on atomic weapons. That was the personal order of I.V. Stalin.

Both seminar of Lyusternik and his own intensive engagement in solving of applied mathematic problems increased interest in computing and -especially- in automation of programming. He and M.K. Gavurin established contacts with Leningrad factory of calculating machines, which conducted simple accounting and statistics, using so called analytic-calculating machines (model of 1939) that were in fact modernised Hollerith tabulators.

Soon they developed method of *parallel simple calculations* what made possible implementation of simple programming performed manually on a switchboard. They worked out method of tables accelerated selection and implemented scalar multiplication with one of multiplicands given in binary form. Kantorovich noticed that calculation of Besselian functions up to 120 decimal places, performed at the Leningrad department of V.A. Steklov Mathematic Institute AS USSR, was their main result of the time. Most exciting part of the work consisted in *parallel calculating* of integrals of differential equations for those functions. Parallelism was achieved by dividing integrated intervals into several sub-intervals. Functions of different numbers were calculated simultaneously on each sub-interval. The method allowed forming of large arrays of similar operations what made their calculation efficient even with those clumsy machines.

Interestingly, the same Besselian functions were also calculated in the USA but with computers MARC and ENIAC. In fact Kantorovich started the work two years later but completed it within eighteen months, before the Americans published their tables [5].

Besides interest in pure mathematics he turned to engineering aspects of computing. An original automatic selector of values for various table functions, “functional converter” was devised soon after. That was rather simple electromechanical structure containing about 10 thousand semiconductor components and providing quite sufficient performance. With form and size it resembled a piano. Several those devices were manufactured by the Moscow Plant of Analytical Machinery (MPAM). Its switchboard was built into it, what permitted manual selection of necessary functions. Each function was realised with special circuit. Kantorovich used parallelism in its work [3]. The “converter” was able for calculating up to 10 functions simultaneously (with lower precession). It was registered (patented) as an invention and used for some time. However it didn’t work long. Soon MPAM was integrated into a powerful centre of electronic computer development – SDB-245, which produced the first Soviet serial computer STRELA. The famous family of digital computers – BESM was simultaneously set up by S.A. Lebedev (Lebedev Sergey Alekseevich ).

Continuing theoretical research on *functional analysis* Kantorovich proved its practical applicability and found efficient ways of usage in solution of applied problems. The next -logically following- step consisted in demonstration of its broad applicability in *computation mathematics*. However, those days it did not seem to be logical at all. Although the mentioned –successful- cycle of works, “Functional Analysis and Applied Mathematics” was published in academy journals “*Uspekhi matematicheskih nauk” *and *“Doklady AN SSSR”* in 1948, the audience first accepted the title as absolutely paradoxical. Kantorovich mentioned that in his “Autobiography”.

His cycle of works on theory of approximate methods of functional analysis – awarded the “Stalin Premium” in 1949, greatly influenced following progress of computational mathematics. That was an event of special importance, because by the 1950-s the first electronic digital computers had been already in use and their development was steadily intensifying.

Kantorovich was one of the first to use *linear programming* as a tool in economic researches. In 1948-1950 he utilised *methods of linear programming* in rational cutting of metal sheets at the Leningrad railway carriage works. In the monograph, “Calculations for rational Cutting of industrial Materials”, published jointly with his collaborator V.A. Zalgaller in 1951, he generalised that experience and systematised algorithms of linear programming. He also implemented *dynamic programming *to solve the problem of rational cutting (independently of R. Bellman) and combined it with algorithms of *linear programming*.

All that time he did not interrupt his own development of computing devices. The few first electronic computers of the 1950-s, mounted in the most important governmental and academic centres, were expensive and unreliable. They could not satisfy rapidly growing needs of “broad scientific circles”, so most of the customers were still using traditional arithmometers or imported desktop calculators, like “Merzedes” or ”Rheinmetall”. Besides of that maintenance of big electronic digital computers was very complicated and expensive.

Kantorovich’s experience with “converter” turned to be very valuable. His team designed and patented (USSR Author’s Certificate of 1958) small “mathematical automata”, which proved to be simple in operation, reliable and efficient. In case of a failure it only needed some simple replacements. That was an interesting model of key input electric relay calculator, which automatically performed arithmetic operations.

Three factories signed manufacturing contracts. One of them, situated in town of Vilnius (capital of Lithuania), made a good model called VILNIUS machine made by another factory was called VYATKA[6]. More then 40,000 of those calculators were produced within ten years. Customers’ demand was more or less satisfied. “At least there was no more need in importing”, remarked Kantorovich. They were described in detail in his article, “Key-input Relay computer for automatic performing arithmetic operations” (1959).

Those machines were reliable and remained in use until the 1970-s, when they were finally replaced with new desktop electronic micro-calculators. “The progress is inevitable”, he commented philosophically.

## Advanced method – “Large-block Programming”

In 1956 he reported on “Functional Analysis and computational Mathematics” at the Third All-Union mathematical Congress, jointly with S.L. Sobolev and L.A. Lyusternik (Lusternik Lasar' Aronovich). Simultaneously he published an article “Perspectives of Development and Utilisation of electronic Computers” in collection of works “Mathematics – its Contents, Methods and Significance”. He analysed in it influence of computers on scientific and engineering progress. He also discussed problems emerging in computational mathematics as result of re-evaluation both of the numeric analysis methods, approximate computations and, in a sense, of mathematical problems and applications in general.

Interestingly analytical conclusions on computing (basically very progressive) chiefly came from his previous working experience with mechanical and electro-mechanical calculators but not with electronic computers. Those facilities, namely tabulators and other computation-analytical machinery, remained the basic equipment at numerous “machine-calculating stations” until -minimum- the beginning of the 1950-s.

His growing interest in programming was quite natural as it could realise his mathematical methods on practice. He had been systematically studying automation of programming since the first electronic computers appeared in scientific organisations (around 1953). Soon after he set up a school of “Large-block Programming” in Leningrad. Its activity was first focused on finding “bridges” to cover the semantic rupture between the input language of computer displaying the programs to be performed and mathematical language describing algorithms of a problem solution. Pioneer method of the *large block programming* became an important domain of his work. Nowadays it is considered to be a form of *functional programming*. Speaking in less formal terms, it is a kind of performing a program in functional language, what means calling of a function where values of other functions stay for its arguments. Each of the latter in its turn could be -in general- a superposition of arbitrary depth.

The first publications on *large-block programming system*, which he humorously named “Construction manager”, appeared in 1956. L.T. Petrova – author of a commentary[7] on the “*Large-block programming method*”, noticed that those very advanced ideas predetermined following development of programming for several decades and basically coincided with its mainstream. Anyway, famous *operator method* of programming created by A.A. Lyapunov in Moscow in the first half of the 1950-s (which included his “Schemas of programs”) undoubtedly belonged to the same scientific domain, what could be an indirect proof of his ideas.

Leningrad school of programming was very active in the period between 1954 and early 1960-s. His independent ideology now is often classified as *functional style**.* He wrote, “Soon after appearance of the first electronic digital computers we in Leningrad joined work on simplification and automation of programming. In my opinion, difference between existing machine language and the descriptive mathematical (algorithmic) language made the biggest disadvantage of existing programming. While mathematics uses integrated operations and various notions machine program needs standard operations with simple numbers”. Description of extended calculation scheme, containing integrated (enlarged) mathematical operations, was his first innovation. Another novelty consisted in implementation of arrays with description of their locations as program components. The whole resulting scheme was written as a compact computing plan with logical links and jumps. Subroutines were devised for enlarged operations such as scalar multiplication, sequencing, matrix operations, etc. Most typical (for the concrete program) operations were selected and assigned special inventory numbers (or tugs) instead of machine addresses. They could be probably named “passports” of the data belonging to a concrete array or an operation. That system received the name “large block programming” (what was in fact a popular term in civil house-building engineering of those days). “Implementation of logical schemes for description of calculations extended our possibilities enabling direct performance of various analytical procedures. In other words we were able conduct analytical differentiation of every complex function composed of the primary and some special functions”.

The first three publications, dated by 1956, briefly described fundamentals of his theory and displayed the first practical applications. Characteristic feature of the large-block schemas consisted in their dealing with integrated *values* that is enlarged (or generalised) *information*** objects**. Those were matrixes, arrays, vectors, sequences, schemas, etc. (but never single numbers). Standard processing programs for their contents (sub-routines) were stored in computer memory and automatically called/performed on lower levels of the

*processing hierarchy.*That was the way of

*hierarchical composing*of programming languages. Basic idea consisted in removing all unnecessary small details from higher levels of a language. According to the schema each value to be put into computer should have three characteristics:

- personal name,
- reference (information about its type and structure and its location in memory – its “certificate”, so to say),
- recording – that’s the value itself plus physical representation of its components and its denotation.

Therefore, already in the beginning Kantorovich considered and specified *syntactic*, *semantic* and *interpretation* levels of information objects accepted in their “integrity”. Computing process itself was represented, as three-dimensional one. It was running either on each level, in turns, or on all three simultaneously. Simplification of computing was reached by rational “jumping” between the levels during data processing. From the very beginning all programs performed matrix transpositions and other conversions on “certificate” level. He also introduced an original method of translation – flexible combination of compiling and interpretation. Rigorous criteria of rationality (“economy”) were imposed on selection of each new procedure, operation, etc. Descriptions of that method were published in 1957.

Outside the *values* Kantorovich introduced one more basic information object – *abstract schemas*. Here he replaced traditional formulas (with braces) with a split scheme representing links and dependencies of processed mathematical objects:

K = K1 K2 … Kn

K1 = …

Here K and Ki are components of some indexed set composed of various mathematical objects with different “nature” (numbers, matrixes, functions, etc.). A scheme displays the system (structure) of their interrelations. Each line written as K = K1 K2 … Kn means that the object K is defined by the system of objects K1, K2, … Kn, and the “K” is called the “result”, while others are defined as “arguments”. Direct offspring connection between the result and the arguments is considered and the notion of ** explicit scheme** is introduced.

Another group of principal information objects introduced by Kantorovich consisted of ** abstract schemas**. They displayed relations between the objects – direct dependences between a result and its arguments were analysed. Also the notion of

**was introduced. Undoubted advantage of**

*explicit schema**abstract schemas*method consisted in possibility of their analysis and transformation on syntactic level. Compatibility and ratio of the arguments consistency, notion of the

*schema solution*and transformations of schemas on schematic identity patterns (conversion according to available “master schemas”) were introduced in the hierarchy and studied on semantic level. “

**” was eventually worked out as a special programming language.**

*Schemata symbolic*Description of computation schema, made in terms of so-called “construction managers”, not only covered simplest arithmetic operations but also included numerous enlarged ones (operations with objects, e.g. scalar products, ordering of arrays, operations with matrixes, etc.).

Many principal solutions proposed in his large-block schemas are still in use. His schemata conception, modelling (multilevel) approach, translation methods, which flexibly combined compiling and interpretation – all of them can be traced in contemporary programming systems. A statement that his work influenced development of programming for more than 30 following years won’t be an exaggeration. Notably, it happened when the programs were only written in machine codes.

In 1968 he published an article presenting analysis of all researches performed by the *school of large-block programming*, with about 40 references. Publications of 1974 were dedicated to systematised complex approach to consolidated development of mathematical modelling and analysis, numerical methods and algorithms, programming and computer design performed on basis of the *large-block programming, *as possible means of their consistency.

However, he complained with inability to complete the work on automatic programming system, chiefly due to shortage of personnel. Concentration of efforts on control operations, caused by increased “machine time hunger” of that time, was another reason. It was better suited for mass problems and not for special ones. A schematic description (sort of schedule) of computation was later utilised in programming of computer MIR-3, created in Kiev by V.M. Glushkov. Still later it was utilised by some physicists in complicated computer assisted analytic researches. At the same time Kantorovich’s team was developing applied programs for linear programming and also for traffic and transportation problems. In 1958, M.A. Yakovleva used his potential method for making very compact and efficient programs.

In 1960 a new Siberian branch of the AS USSR was established in Novosibirsk. New scientific centre – large satellite district Akademgorodok, was specially constructed for that purpose. Kantorovich moved there to hold an academic chair of mathematics and economics for about ten next years (1961-1971).

## More computers - unusual project of Arithmetic Machine[8].

Besides microcomputers Kantorovich also designed a bigger machine. That took place in the 1963-1965, when he was already working in Novosibirsk. He created a high-performance computing system “Arithmetic Machine”, or “computer AM”, with efficient architecture. In reality that was specialised processor or rather “computation compound” of a universal computer and a minicomputer. It was described in his work, “Computation System Consisting of Universal Digital Computing Machine and Minicomputer” (1965).

His basic idea consisted in higher specialisation of each arithmetic unit. Operation of that machine (AM) was based on Kantorovich’s “rotor” principle of “mass calculation operations performance”. The operations were performed with maximal speed limited only with operation storage speed. According to the project its structure corresponded to a multi-level pipeline with direct memory access, followed by an original multi-input carry-saving adder. Design of this pipeline computer received several patents (USSR inventor’s certificates). Some architectural solutions embedded in AM were later widely implemented both in the Soviet and foreign computers [6].

Generally speaking, *utilisation of problem-oriented processors* is considered to be one of the most perspective directions of contemporary computer development.

In 1969 completed pilot model was already in operation at the Novosibirsk Academic computer centre. However, AM never entered commercial production.

Nevertheless he didn’t give up and continued the work as theoretical research. In 1974 he published work, “Complex Approach to Performance of mass Computations”, where he set forth and reasoned ideas on complex development of computational mathematics (“machine mathematics”) – methods, algorithms, programming, structure of machines.

In 1971-1976 he moved back to Moscow to continue his scientific work and supervise researches at the Institute of Control in National Economy.

In the 1970-s Kantorovich suggested ideas on improvement of various decimal computing devices, in his works “A Device for Multiplication” (1973) and “Electromechanical Storage Device” (1974). He also worked on *automation of programming*, as well as of some other forms of human intellectual activity (such as symbolic notation, transforming of programs, etc.). The principles, he described in the work “About a Mathematical Symbolic Convenient at Performing Calculations on Computers” (1977), were subsequently developed by some Soviet and foreign authors.

Leonid Vitalevich Kantorovich died on the 7 of April 1986 and is buried on the central cemetery of Moscow.

## Awards and honours

Besides higher scientific titles and premiums L.V. Kantorovich was decorated with two orders “Lenin Order”, three orders “Red Banner of Labour”, order “Sign of Honour” and medals. He was elected by many foreign academies and scientific societies “Honoris causa”. He was –also “honoris causa”- a doctor of the universities of Glasgow, Warsaw, Grenoble, Nice, Munich, Helsinki, Paris (Sorbonne), Cambridge, Statistical University in Calcutta, Pennsylvania.

Nowadays ** linear programming**is a widely studied subject. It is usually described in school and university textbooks. Its methods are integral part of every modern economic analysis and they are included in, constantly developing, applied software.

Notes

1. very good photo gallery on L.V. Kantorovich can be found in his “family album”: ttp://www.kantorovich.icape.ru/album.html

2. B.N. Delone – Soviet mathematician – http://www-history.mcs.st-and.ac.uk/Biographies/Delone.html

*(**Russ**.) Борис Николаевич ДЕЛОНЕ* (1890-1980) http://higeom.math.msu.su/history/delone_r.html

3. V. Zalgaller – a collaborator of L.V. Kantorovich; (in article) “About remarkable person – Leonid Vitalyevich Kantorovich”, (Russ. *О ЗАМЕЧАТЕЛЬНОМ ЧЕЛОВЕКЕ — ЛЕОНИДЕ ВИТАЛЬЕВИЧЕ КАНТОРОВИЧЕ ) *http://www.peoples.ru/science/mathematics/kantorovich/

4. During the whole war period those who had doctor’s degree and/or professorship were not subjected to call to military service as persons of importance for maintaining defence industry, corresponding scientific researches and education of specialists.

5. the given quotation is not exact, since the source text had been previously translated and re-translated several times, however the main idea was not distorted (A.N.).

6. named after a town in the West Urals

7. L.T. Petrova. Commentaries on Kantorovich’s “Large-block programming method”. // In: Essays on history of Computer Science in Russia. D.A. Pospelov, J.I. Fet. 1998, pp.446-448.

8. This section is chiefly based on materials given in the source [6].

Sources:

- “Leonid Vitalevich Kantorovich – an academician” (
*Леонид**Витальевич**Канторович**Академик*) – an article in “Russian Virtual Computer Museum”

http://www.computer-museum.ru/galglory/21.htm -
**G. Trogemann, A.Y. Nitussov, W. Ernst (Eds.).**“Computing in Russia” (Engl. VIEWEG, Wiesbaden, 2001, p. 350). -
**V. Zalgaller.**“About a remarkable person – Leonid Vitalyevich Kantorovich” (Russ.)

http://www.vestnik.com/issues/2003/0820/win/zalgaller.htm - Kantorovich Leonid in, “The MacTutor History of Mathematics archive”

http://www-history.mcs.st-and.ac.uk/Mathematicians/Kantorovich.html - http://www.kantorovich.icape.ru/autors.html (Russian archive of L.V. Kantorovich).
- “L.V. Kantorovich – Mathematician and Economist”,
**S.S. Kutaladze, V.L. Makarov, I.V. Romanovskiy, G.Sh. Rubinshtein**(Russ.)

http://www.mathsoc.spb.ru/pantheon/kantorov/lvkumn.html - http://www.cemi.rssi.ru/rus/persons/kantorov.htm
- http://www.math.nsc.ru/LBRT/u2/nauka/kant.html

19.01.2012