Vitold Belevitch (2 March 1921 – 26 December 1999) was a Belgian mathematician and electrical engineer of Russian origin who produced some important work in the field of electrical network theory. Born to parents fleeing the

Bolshevik The Bolsheviks (russian: Большевики́, from большинство́ ''bol'shinstvó'', 'majority'),; derived from ''bol'shinstvó'' (большинство́), "majority", literally meaning "one of the majority". also known in English ...

s, he settled in Belgium where he worked on early computer construction projects. Belevitch is responsible for a number of circuit theorems and introduced the now well-known

scattering parameters Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals. The parameters are useful f ...

. Belevitch had an interest in languages and found a mathematical derivation of Zipf's law. He also published on machine languages. Another field of interest was transmission lines, where he published on line coupling. He worked on telephone conferencing and introduced the mathematical construct of the conference matrix.

Early life

Belevitch was born 2 March 1921 in Terijoki,

Karelia Karelia ( Karelian and fi, Karjala, ; rus, Каре́лия, links=y, r=Karélija, p=kɐˈrʲelʲɪjə, historically ''Korjela''; sv, Karelen), the land of the Karelian people, is an area in Northern Europe of historical significance for ...

, now incorporated into Russia, but at the time part of Finland. Belevitch's parents were Russian and his mother was an ethnic Pole. They were attempting to flee from their home in Petrograd ( St Petersburg) in Russia to escape the

Bolshevik revolution The October Revolution,. officially known as the Great October Socialist Revolution. in the Soviet Union, also known as the Bolshevik Revolution, was a revolution in Russia led by the Bolshevik Party of Vladimir Lenin that was a key mome ...

, which Belevitch's father opposed. Belevitch's heavily pregnant mother succeeded in crossing the border into Finland and continued on to

Helsinki Helsinki ( or ; ; sv, Helsingfors, ) is the Capital city, capital, primate city, primate, and List of cities and towns in Finland, most populous city of Finland. Located on the shore of the Gulf of Finland, it is the seat of the region of U ...

after Vitold was born, where the birth was registered. She headed for Helsinki because her husband's father was principal of the Russian school there. Belevitch's father was arrested before he could follow and was deported to

Siberia Siberia ( ; rus, Сибирь, r=Sibir', p=sʲɪˈbʲirʲ, a=Ru-Сибирь.ogg) is an extensive geographical region, constituting all of North Asia, from the Ural Mountains in the west to the Pacific Ocean in the east. It has been a part o ...

, where he died without ever seeing his son. In 1926 Belevitch, while still a small child, emigrated with his mother to Belgium.

Education

Belevitch was educated in French in Belgium, until July 1936 at the Notre-Dame de la Paix College at

Namur Namur (; ; nl, Namen ; wa, Nameur) is a city and municipality in Wallonia, Belgium. It is both the capital of the province of Namur and of Wallonia, hosting the Parliament of Wallonia, the Government of Wallonia and its administration. Na ...

. In 1937, aged 16, he enrolled at the

Université Catholique de Louvain The Université catholique de Louvain (also known as the Catholic University of Louvain, the English translation of its French name, and the University of Louvain, its official English name) is Belgium's largest French-speaking university. It ...

where he studied electrical and mechanical engineering, graduating in 1942. Belevitch gained his PhD in applied sciences at the same university in 1945. His sponsor was

Charles Lambert Manneback Charles Lambert Marie Joseph Manneback (born 9 March 1894 in Etterbeek, Belgium; died 15 December 1975 in Etterbeek) was a Belgian physicist, mining engineer, and mathematician.

and his second advisor was

Wilhelm Cauer Wilhelm Cauer (24 June 1900 – 22 April 1945) was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters and his work marked the beginning of the field of network synthesis. Prior ...

, the founder of the field of network synthesis. From 1953 until 1985 Belevitch lectured at the university. He taught

circuit theory Circuit may refer to: Science and technology Electrical engineering * Electrical circuit, a complete electrical network with a closed-loop giving a return path for current ** Analog circuit, uses continuous signal levels ** Balanced circ ...

and other mathematical subjects related to electrical science. In 1960 he became a special professor (buitengewoon hoogleraar). Although Belevitch worked as an electrical engineer, his primary interest was mathematics, especially algebra. There was a tradition in Belgium of the most gifted mathematicians entering engineering rather than pure mathematics or physics. Belevitch showed his mathematical leanings by preferring the use of blackboard and chalk to any

audio-visual Audiovisual (AV) is electronic media possessing both a sound and a visual component, such as slide-tape presentations, films, television programs, corporate conferencing, church services, and live theater productions. Audiovisual service p ...

aids during lectures. He even lectured in this way when presenting the opening lecture to a large audience at an international conference at the IEE in London.

Career

After graduating in 1942, Belevitch joined the Bell Telephone Manufacturing Company (BTMC) in Antwerp, originally part of the International Bell Telephone Company headquartered in

Brussels Brussels (french: Bruxelles or ; nl, Brussel ), officially the Brussels-Capital Region (All text and all but one graphic show the English name as Brussels-Capital Region.) (french: link=no, Région de Bruxelles-Capitale; nl, link=no, Bruss ...

but, along with their other European holdings, sold to International Telephone and Telegraph (ITT) in 1925. At BTMC Belevitch became head of the transmission department. It was here that he came into contact with Wilhelm Cauer who became a great influence on him. Cauer was one of the leading circuit theorists of the day and at the time worked for

Mix & Genest Mix & Genest was founded on 1 October 1879 by the businessman Wilhelm Mix and the engineer Werner Genest in Berlin-Schöneberg. The company was initially an 1879 branch of the ITT Corporation. It was very successful and became one of the pioneers ...

Berlin Berlin is Capital of Germany, the capital and largest city of Germany, both by area and List of cities in Germany by population, by population. Its more than 3.85 million inhabitants make it the European Union's List of cities in the European U ...

, a sister company under the ITT umbrella. Cauer died during the

Second World War World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

but Belevitch long after continued to consider his works to be the highest authority on matters of circuit theory. From 1951 Belevitch was involved in the design of electronic computers which BTMC were developing for the Belgian government. The purpose of this program was to "catch up" with the advances made in the English-speaking world during the war. It resulted in the construction of the Machine mathématique IRSIA-FNRS.IRSIA-FRNS: named after the two Belgian government departments sponsoring the project.
IRSIA: ''Institut pour l’Encouragement de la Recherche Scientifique dans l’Industrie et dans l’Agriculture'' (Institute for Promotion of Scientific Research in Industry and Agriculture)
FNRS: ''Fonds National de la Recherche Scientifique'' (National Fund for Scientific Research) From 1952 Belevitch represented the electrical engineering aspect of this project. In 1955 Belevitch became director of the Belgian Computing Centre (Comité d'Étude et d'Exploitation des Calculateurs Électroniques) in Brussels which operated this computer for the government. Initially, only the 17-rack prototype was operational. One of the first tasks to which it was put was the calculation of

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

s. The full 34-rack machine was moved from Antwerp and put into service in 1957. Belevitch used this machine to investigate

transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed alg ...

s. In 1963 Belevitch became head of the newly formed Laboratoire de Recherche MBLEMBLE: ''Manufacture Belge de Lampes Electriques'', a company originally selling light bulbs under the

Mazda , commonly referred to as simply Mazda, is a Japanese Multinational corporation, multinational Automotive industry, automotive manufacturer headquartered in Fuchū, Hiroshima (town), Fuchū, Hiroshima Prefecture, Hiroshima, Japan. In 2015, M ...

brand, later entirely absorbed into Philips. See MBLE on

French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...

or Dutch Wikipedia. (later Philips Research Laboratories Belgium) under the

Philips Koninklijke Philips N.V. (), commonly shortened to Philips, is a Dutch multinational conglomerate corporation that was founded in Eindhoven in 1891. Since 1997, it has been mostly headquartered in Amsterdam, though the Benelux headquarters is ...

director of research Hendrik Casimir in

Eindhoven Eindhoven () is a city and municipality in the Netherlands, located in the southern province of North Brabant of which it is its largest. With a population of 238,326 on 1 January 2022,filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component th ...

modulator In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...

s, coupled lines, and non-linear circuits. He was on the editorial board of the ''International Journal of Circuit Theory'' from its foundation in 1973. He also made major contributions in

information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...

, electronic computers, mathematics and

linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Lingu ...

. Belevitch dominated international conferences and was prone to asking searching questions of the presenters of papers, often causing them some discomfort. The organiser of one conference at

Birmingham University The University of Birmingham (informally Birmingham University) is a Public university, public research university located in Edgbaston, Birmingham, United Kingdom. It received its royal charter in 1900 as a successor to Queen's College, Birmingha ...

in 1959 made Belevitch the chairman of the session in which the organiser gave his own presentation. It seems he did this to restrain Belevitch from asking questions. Belevitch stopped attending conferences in the mid-1970s with the exception of the IEEE International Symposium on Circuits and Systems in

Montreal Montreal ( ; officially Montréal, ) is the second-most populous city in Canada and most populous city in the Canadian province of Quebec. Founded in 1642 as '' Ville-Marie'', or "City of Mary", it is named after Mount Royal, the triple- ...

in 1984 in order to receive the IEEE Centennial Medal.

Circuit theory

Scattering matrix

It was in his 1945 dissertation that Belevitch first introduced the important idea of the scattering matrixIt is now recognised that the

S-parameters Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals. The parameters are useful f ...

used in circuit theory are in all essentials comparable with the scattering matrices used by physicists in

scattering theory In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance su ...

to describe the

scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...

behaviour of particles and waves. The "S" in S-parameter stands for ''scattering'' in recognition of this. (called ''repartition matrix'' by Belevitch). This work was reproduced in part in a later paper by Belevitch, ''Transmission Losses in 2''n''-terminal Networks''. Belgium was occupied by

Nazi Germany Nazi Germany (lit. "National Socialist State"), ' (lit. "Nazi State") for short; also ' (lit. "National Socialist Germany") (officially known as the German Reich from 1933 until 1943, and the Greater German Reich from 1943 to 1945) was ...

for most of

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

and this prevented Belevitch from any communication with American colleagues. It was only after the war that it was discovered that the same idea, under the ''scattering matrix'' name, had independently been used by American scientists developing military

radar Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...

s. The American work by Montgomery,

Dicke Dicke is a surname. Notable people with the surname include: * Amie Dicke (born 1978), Dutch artist * Finn Dicke (born 2004), Dutch footballer * Pien Dicke (born 1999), Dutch field hockey player * Robert H. Dicke (1916–1997), American physicist ...

and

Purcell Henry Purcell (, rare: September 1659 – 21 November 1695) was an English composer. Purcell's style of Baroque music was uniquely English, although it incorporated Italian and French elements. Generally considered among the greatest E ...

was published in 1948. Belevitch in his work had applied scattering matrices to lumped-element circuits and was certainly the first to do so, whereas the Americans were concerned with the

distributed-element : ''This article is an example from the domain of electrical systems, which is a special case of the more general distributed-parameter systems.'' In electrical engineering, the distributed-element model or transmission-line model of electrical ...

circuits used at

microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequency, frequencies between 300 MHz and 300 GHz respectively. Different sources define different fre ...

frequencies in radar. Belevitch produced a textbook, ''Classical Network Theory'', first published in 1968 which comprehensively covered the field of passive

one-port In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, as a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside ...

, and multiport circuits. In this work he made extensive use of the now-established S parameters from the scattering matrix concept, thus succeeding in welding the field into a coherent whole. The eponymous

Belevitch's theorem Belevitch's theorem is a theorem in electrical network analysis due to the Russo-Belgian mathematician Vitold Belevitch (1921–1999). The theorem provides a test for a given S-matrix to determine whether or not it can be constructed as a lossless ...

, explained in this book, provides a method of determining whether or not it is possible to construct a passive, lossless circuit from discrete elements (that is, a circuit consisting only of

inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a ...

s and

capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a ...

s) that represents a given scattering matrix.

Telephone conferencing

Belevitch introduced the mathematical concept of conference matrices in 1950, so called because they originally arose in connection with a problem Belevitch was working on concerning telephone conferencing. However, they have applications in a range of other fields as well as being of interest to pure mathematics. Belevitch was studying setting up telephone conferencing by connecting together ideal transformers. It turns out that a necessary condition for setting up a conference with ''n'' telephone ports and ideal signal loss is the existence of an ''n''×''n'' conference matrix. ''Ideal signal loss'' means the loss is only that due to splitting the signal between conference subscribers – there is no dissipation within the conference network. The existence of conference matrices is not a trivial question; they do not exist for all values of ''n''. Values of ''n'' for which they exist are always of the form 4''k''+2 (''k'' integer) but this is not, by itself, a sufficient condition. Conference matrices exist for ''n'' of 2, 6, 10, 14, 18, 26, 30, 38 and 42. They do not exist for ''n'' of 22 or 34. Belevitch obtained complete solutions for all ''n'' up to 38 and also noted that ''n''=66 had multiple solutions.

Other work on circuits

Belevitch wrote a comprehensive summary of the history of circuit theory. He also had an interest in transmission lines, and published several papers on the subject. They include papers on skin effects and coupling between lines ("crosstalk") due to asymmetry. Belevitch first introduced the ''great factorization theorem'' in which he gives a factorization of paraunitary matrices. Paraunitary matrices occur in the construction of filter banks used in multirate digital systems. Apparently, Belevitch's work is obscure and difficult to understand. A much more frequently cited version of this theorem was later published by P. P. Vaidyanathan.

Linguistics

Belevitch was educated in French but continued to speak Russian to his mother until she died. In fact, he was able to speak many languages, and could read even more. He studied

Sanskrit Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had Trans-cul ...

and the etymology of

Indo-European languages The Indo-European languages are a language family native to the overwhelming majority of Europe, the Iranian plateau, and the northern Indian subcontinent. Some European languages of this family, English, French, Portuguese, Russian, ...

. Belevitch wrote a book on human and machine languages in which he explored the idea of applying the mathematics of

to obtain results regarding human languages. The book highlighted the difficulties for machine understanding of language for which there was some naive enthusiasm amongst cybernetics researchers in the 1950s. Belevitch also wrote a paper, ''On the Statistical Laws of Linguistic Distribution'', which gives a derivation for the well-known

empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...

relationship, Zipf's law. This law, and the more complex Mandelbrot law, provide a relationship between the frequency of word occurrence in languages and the word's

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...

. In the simplest form of Zipf's law, frequency is inversely proportional to rank. Belevitch expressed a large class of

statistical distribution In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution functio ...

s (not only the

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...

) in terms of rank and then expanded each expression into a

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...

. In every case Belevitch obtained the remarkable result that a first order truncation of the series resulted in Zipf's law. Further, a second-order truncation of the Taylor series resulted in Mandelbrot's law. This gives some insight into the reason why Zipf's law has been found experimentally to hold in such a wide variety of languages.

Control systems

Belevitch played a part in developing a mathematical test for determining the

controllability Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. Controllability and observabi ...

of linear

control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial ...

s. A system is controllable if it can be moved from one state to another through the system

state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the t ...

in a finite time by application of control inputs. This test is known as the Popov-Belevitch-Hautus, or PBH, test. There is also a PBH test for determining the

observability Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. In control theory, the observability and controllability of a linear system are mathematical duals. The concept of observ ...

of a system – that is, the ability to determine the state of a system in finite time solely from the system's own outputs. The PBH test was originally discovered by Elmer G. Gilbert in 1963, but Gilbert's version only applied to systems that could be represented by a

diagonalizable matrix In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...

. The test was subsequently generalised by Vasile M. Popov (in 1966), Belevitch (in ''Classical Network Theory'', 1968) and

Malo Hautus Mal, which in Spanish means ''bad or evil'', may also refer to: Places *Malo, Italy, a town *Malo Island, formerly known as St. Bartholomew, Vanuatu * Malo (Solomon Islands), an island *Malo, Washington, Ferry County, Washington, United States * ...

in 1969.

IEEE and honours

Belevitch was a Fellow of the

Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operat ...

(IEEE) and was vice-chair of the

Benelux The Benelux Union ( nl, Benelux Unie; french: Union Benelux; lb, Benelux-Unioun), also known as simply Benelux, is a Political union, politico-economic union and formal international intergovernmental cooperation of three neighboring states in ...

section when it was formed in 1959. He was awarded the

IEEE Centennial Medal The IEEE Centennial Medal was a medal minted and awarded in 1984 ''to persons deserving of special recognition for extraordinary achievement'' to celebrate the Centennial of the founding of the Institute of Electrical and Electronics Engineers (I ...

, and in 1993, the Society Award (now called Mac Van Valkenburg Award) of the IEEE Circuits and Systems Society. He was also a member of the North Rhine-Westphalian Academy of Sciences. Belevitch received an honorary doctoral degree in 1975 from the

Technical University of Munich The Technical University of Munich (TUM or TU Munich; german: Technische Universität München) is a public research university in Munich, Germany. It specializes in engineering, technology, medicine, and applied and natural sciences. Establis ...

, and another from the

École Polytechnique Fédérale de Lausanne École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine The Seine ( , ) is a river in northern Franc ...

, Switzerland, in 1978. He was also rewarded with Belgian royal medals. Since 2003, the IEEE Circuits and Systems Society has instituted a Vitold Belevitch award for work in circuit theory. The award is presented biennially at the ''European Conference on Circuit Theory & Design''.

Selected publications

Belevitch was a prolific publisher with around 4000 pages of scientific output. He was publishing throughout his career right up to, and beyond, his retirement in 1984.Vandewalle, p. 429

Books

*''Langage des Machines et Langage Humain'', Bruxelles: Office de publicité, 1956 (in French). *''Théorie des Circuits de Télécommunication'', Louvain: Librairie Universitaire, 1957 (in French). *''Théorie des Circuits Nonlinéaires en Régime Alternatif: Redresseurs, Modulateurs, Oscillateurs'', Louvain: Uystpruyst, 1959 (in French). *''Classical Network Theory'', San Francisco: Holden-Day, 1968 .

Journal articles

"Extension of Norton's method of impedance transformation to band-pass filters"
''Electrical Communication'', vol.24, no.1, pp. 59–65, March 1947.
"Insertion loss and effective phase shift in composite filters at cut-off frequencies"
''Electrical Communication'', vol.24, no.2, pp. 192–194, June 1947.
"Transmission losses in 2''n''-terminal networks"
''Journal of Applied Physics'', vol.19, No. 7, pp. 636–638, 1948.
"Development in the design of filters"
''Electrical Communication'', vol.26, no.1, pp. 84–98, March 1949.
"Theory of 2''n''-terminal networks with application to conference telephony"
''Electrical Communication'', vol.27, no.3, pp. 231–244, September 1950.
"Topics in the design of insertion loss filters"
''IRE Transactions on Circuit Theory'', vol.2, iss.4, pp. 337–346, December 1955.
"Théorie de l'information et statistique linguistique"
''Académie Royale de Belgique: Bulletin de la Classe des Sciences'', series 5, vol.42, pp. 419–436, 1956. (in French) *"On the statistical laws of linguistic distribution", ''Annales de la Société Scientifique de Bruxelles'', vol.3, iss.73, pp. 310–326, 1959 . (in French)
"Summary of the history of circuit theory"
''Proceedings of the IRE'', vol.50, iss.5, pp. 848–855, May 1962. *"Factorization of scattering matrices with applications to passive network synthesis", ''Philips Research Reports'', vol.18, pp. 275–317, 1963 . *"Interpolation matrices", ''Philips Research Reports'', vol.25, pp. 337–369, 1970 . *"On the realizability of non-rational positive real functions", ''International Journal of Circuit Theory and Applications'', vol.1, iss.1, pp. 17–30, March 1973. *"Theory of the proximity effect in multiwire cables", ''Philips Research Reports'', vol.32, pp. 16–43, 96-177, 1977 .
"The Gauss hypergeometric ratio as a positive real function"
''

SIAM Thailand ( ), historically known as Siam () and officially the Kingdom of Thailand, is a country in Southeast Asia, located at the centre of the Indochinese Peninsula, spanning , with a population of almost 70 million. The country is bo ...

Journal on Mathematical Analysis'', vol.13, pp. 1024–1040, 1982. *"Ideal rectifier ''n''-ports are solvable", ''International Journal of Circuit Theory and Applications'', vol.15, iss.2, pp. 171–180, April 1987.

Notes

References

Bibliography

*Belevitch, Vitol
"Summary of the history of circuit theory"
''Proceedings of the IRE'', vol.50, iss.5, pp. 848–855, May 1962. *Chen, Ben M.; Lin, Zongli; Shamash, Yacov ''Linear Systems Theory: a Structural Decomposition Approach'', Boston: Birkhäuser, 2004 . *Colbourn, Charles J.; Dinitz, Jeffrey H. ''Handbook of Combinatorial Designs'', Boca Raton, Florida: Chapman and Hall/CRC Press, 2007 . *Dutoit, Thierry (ed

''Théorie des Circuits et Traitement du Signal Lab'', accessed an

21 May 2011. *Fettweis, Alfre
"In memoriam, Vitold Belevitch"
''IEEE Transactions on Circuits and Systems'', vol.47, iss.5, pp. 613–614, May 2000. *Fettweis, Alfred (in Dutoit) "Reminiscing Vitold Belevitch (1921–1999)", ''European Conference on Circuit Theory and Design 2001'', Plenary session, Helsinki. *Huurdeman, Anton A. ''The Worldwide History of Telecommunications'', New Jersey: Wiley-IEEE, 2003 . *Kisačanin, Branislav; Agarwal, Gyan C. ''Linear Control Systems'', New York: Kluwer Academic/Plenum Publishers, 2002 . *van Lint, Jacobus Hendricus; Wilson, Richard Michael ''A Course in Combinatorics'', Cambridge: Cambridge University Press, 2001 . *Mols, Sandra; d'Udekem-Gevers, Mari
"Disseminating electronics: Bell Telephone and the emergence of electronic computing expertise in post-war Belgium, c.1945–c.1960"
''History of Telecommunications Conference, 2008'', pp. 102–109 Paris: 11–12 September 2008 . *Montgomery, Carol Gray; Dicke, Robert Henry; Purcell, Edward M
''Principles of microwave circuits''
London: Peter Peregrinus Ltd., 1987 (1948 edition is New York: McGraw-Hill, ). *Rockmore, Daniel Nahum; Healy, Dennis M. ''Modern Signal Processing'', Cambridge: Cambridge University Press, 2004 . *Strang, Gilbert; Nguyen, Truong ''Wavelets and Filter Banks'', Wellesley, MA: Wellesley-Cambridge Press 1996 . *Stinson, Douglas Robert ''Combinatorial Designs: Constructions and Analysis'', New York: Springer, 2004 . *Valkenburg, Mac Elwyn Van ''Circuit Theory: Foundations and Classical Contributions'', Stroudsburg, Pennsylvania: Dowden, Hutchinson & Ross, 1974 . *Vaidyanathan, P. P. ''Multirate systems and filter banks'', Englewood Cliffs, NJ: PTR Prentice Hall, 1993 . *Vandewalle, Joos "In memoriam – Vitold Belevitch", ''International Journal of Circuit Theory and Applications'', vol.28, iss.5, pp. 429–430, September/October 2000. *de Vries, Marc; Boersma, Kees ''80 years of research at the Philips Natuurkundig Laboratorium (1914-1994)'', Amsterdam University Press, 2005 . *Walker, Mark ''Science and Ideology: a Comparative History'', Routledge, 2003 .