Aizik Volpert

Aizik Isaakovich Vol'pert
Born	(1923-06-05)5 June 1923[1][2] Kharkov
Died	January 2006 (2006-02) (aged 82) Haifa
Alma mater	Lviv University Moscow University[3]
Known for	Functions of bounded variation index theory partial differential equations parabolic partial differential equations mathematical chemistry
Scientific career
Institutions	Lviv University Lviv Industrial Forestry Institute Technion

Aizik Isaakovich Vol'pert (Russian: Айзик Исаакович Вольперт) (5 June 1923[1][2] – January 2006) (the family name is also transliterated as Volpert[4] or Wolpert[5]) was a Soviet and Israeli mathematician and chemical engineer[6] working in partial differential equations, functions of bounded variation and chemical kinetics.

Life and academic career

The Programme Committee of the Russian Conference "Mathematical Methods in Chemical Kinetics", Shushenskoye, Krasnoyarsk Krai, 1980. From left to right: A.I. Volpert, V.I. Bykov, A.N. Gorban, G.S. Yablonsky, A.N.Ivanova.

Vol'pert graduated from Lviv University in 1951, earning the candidate of science degree and the docent title respectively in 1954 and 1956 from the same university:[1] from 1951 on he worked at the Lviv Industrial Forestry Institute.[1] In 1961 he became senior research fellow[7] while 1962 he earned the "doktor nauk"[2] degree from Moscow State University. In the 1970s–1980s A. I. Volpert became one of the leaders of the Russian Mathematical Chemistry scientific community.[8] He finally joined Technion’s Faculty of Mathematics in 1993,[3] doing his Aliyah in 1994.[9]

Work

Index theory and elliptic boundary problems

Vol'pert developed an effective algorithm for calculating the index of an elliptic problem before the Atiyah-Singer index theorem appeared:[10] He was also the first to show that the index of a singular matrix operator can be different from zero.[11]

Functions of bounded variation

He was one of the leading contributors to the theory of BV-functions: he introduced the concept of functional superposition, which enabled him to construct a calculus for such functions and applying it in the theory of partial differential equations.[12] Precisely, given a continuously differentiable function f : ℝ^p → ℝ and a function of bounded variation u(x) = (u₁(x),...,u_p(x)) with x ∈ ℝ^n and n ≥ 1, he proves that f∘u(x) = f(u(x)) is again a function of bounded variation and the following chain rule formula holds:[13]

${\displaystyle {\frac {\partial f({\boldsymbol {u}}({\boldsymbol {x}}))}{\partial x_{i}}}=\sum _{k=1}^{p}{\frac {\partial {\bar {f}}({\boldsymbol {u}}({\boldsymbol {x}}))}{\partial u_{k}}}{\frac {\partial {u_{k}({\boldsymbol {x}})}}{\partial x_{i}}}\qquad \forall i=1,\ldots ,n}$

where –f(u(x)) is the already cited functional superposition of f and u. By using his results, it is easy to prove that functions of bounded variation form an algebra of discontinuous functions: in particular, using his calculus for n = 1, it is possible to define the product H ⋅ δ of the Heaviside step function H(x) and the Dirac distribution δ(x) in one variable.[14]

Chemical kinetics

His work on chemical kinetics and chemical engineering led him to define and study differential equations on graphs.[15]

Selected publications

• Hudjaev, Sergei Ivanovich; Vol'pert, Aizik Isaakovich (1985), Analysis in classes of discontinuous functions and equations of mathematical physics, Mechanics: analysis, vol. 8, Dordrecht–Boston–Lancaster: Martinus Nijhoff Publishers, pp. xviii+678, ISBN 90-247-3109-7, MR 0785938, Zbl 0564.46025. One of the best books about BV-functions and their application to problems of mathematical physics, particularly chemical kinetics.
• Vol'pert, Aizik Isaakovich (1967), Пространства BV и квазилинейные уравнени, Matematicheskii Sbornik, (N.S.) (in Russian), 73(115) (2): 255–302, MR 0216338, Zbl 0168.07402. A seminal paper where Caccioppoli sets and BV functions are thoroughly studied and the concept of functional superposition is introduced and applied to the theory of partial differential equations: it was also translated as Vol'Pert, A. I. (1967), "Spaces BV and quasi-linear equations", Mathematics of the USSR-Sbornik, 2 (2): 225–267, Bibcode:1967SbMat...2..225V, doi:10.1070/SM1967v002n02ABEH002340, hdl:10338.dmlcz/102500, MR 0216338, Zbl 0168.07402.
• Vol'pert, Aizik Isaakovich (1972), Дифференциальные уравнения на графах, Matematicheskii Sbornik, (N.S.) (in Russian), 88(130) (4(8)): 578–588, MR 0316796, Zbl 0242.35015, translated in English as Vol'Pert, A. I. (1972), "Differential equations on graphs", Mathematics of the USSR-Sbornik, 17 (4): 571–582, Bibcode:1972SbMat..17..571V, doi:10.1070/SM1972v017n04ABEH001603, Zbl 0255.35013.
• Vasiliev, V. M.; Volpert, A. I.; Hudiaev, S. I. (1973), "On the method of quasi-stationary concentrations for chemical kinetics equations", Журнал вычислительной математики и математической физики (in Russian), 13 (3): 683–697.
• Vol'pert, A. I. (1976), "Qualitative methods of investigation of equations of chemical kinetics", Preprint (in Russian), Institute of Chemical Physics, Chernogolovka.
• Vol'pert, V. A.; Vol'pert, A. I.; Merzhanov, A. G. (1982), "Application of the theory of bifurcations in study of the spinning combustion waves", Doklady Akademii Nauk SSSR (in Russian), 262 (3): 642–645.
• Vol'pert, V. A.; Vol'pert, A. I.; Merzhanov, A. G. (1982b), "Analysis of nonunidimensional combustion modes by bifurcation theory methods", Doklady Akademii Nauk SSSR (in Russian), 263 (4): 918–921.
• Vol'pert, V. A.; Vol'pert, A. I.; Merzhanov, A. G. (1983), "Application of the theory of bifurcations to the study of unsteady regimes of combustion", Fizika Goreniya i Vzryva (in Russian), 19: 69–72, translated in English as Vol'Pert, V. A.; Vol'Pert, A. I.; Merzhanov, A. G. (1983), "Application of the theory of bifurcations to the investigation of nonstationary combustion regimes", Combustion, Explosion, and Shock Waves, 19 (4): 435–438, Bibcode:1983CESW...19..435V, doi:10.1007/BF00783642, S2CID 97950149.
• Vol'pert, V. A.; Vol'pert, A. I. (1989), "Existence and stability of traveling waves in chemical kinetics", Dynamics of Chemical and Biological Systems (in Russian), Novosibirsk: Nauka, pp. 56–131.
• Vol'pert, Aizik I.; Vol'pert, Vitaly A.; Vol'pert, Vladimir A. (1994), Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140, Providence, R.I.: American Mathematical Society, pp. xii+448, ISBN 0-8218-3393-6, MR 1297766, Zbl 1001.35060.
• Vol'pert, A. I. (1996), "Propagation of Waves Described by Nonlinear Parabolic Equations (a commentary on article 6)", in Oleinik, O. A. (ed.), I. G. Petrovsky Selected works. Part II: Differential equations and probability theory, Classics of Soviet Mathematics, vol. 5 (part 2), Amsterdam: Gordon and Breach Publishers, pp. 364–399, ISBN 2-88124-979-5, MR 1677648, Zbl 0948.01043.
• Vol'pert, V. A.; Vol'pert, A. I. (1998), "Convective instability of reaction fronts: linear stability analysis", European Journal of Applied Mathematics, 9 (5): 507–525, doi:10.1017/S095679259800357X, MR 1662311, S2CID 121533943, Zbl 0918.76027.

See also

• Atiyah-Singer index theorem
• Bounded variation
• Caccioppoli set
• Differential equation on a graph

Notes

1. See Kurosh et al. (1959b, p. 145).
2. See Fomin & Shilov (1969, p. 265).
3. According to the few information given by the Editorial staff of Focus (2003, p. 9).
4. See Chuyko (2009, p. 79).
5. See Mikhlin & Prössdorf 1986, p. 369.
6. His training as an engineer is clearly indicated by Truesdell (1991, p. 88, footnote 1) who, referring to the book (Hudjaev & Vol'pert 1985), writes exactly:-"Be it noted that this clear, excellent, and compact book is written by and for engineers".
7. Precisely he became "старший научный сотрудник", abbreviated as "ст. науч. сотр.", according to Fomin & Shilov (1969, p. 265).
8. Manelis & Aldoshin (2005, pp. 7–8) detail briefly Vol'pert's and other scientists contribution to the development of mathematical chemistry. Precisely, they write that "В работах математического отдел института ( А. Я. Повзнер, А. И. Вольперт, А. Я. Дубовицкий) получили широкое развитие математической основи химической физики: теория систем дифференциальных уравнений, методы оптимизации, современные вычислительные методы методы отображения и т.д., которые легли в основу современной химической физики (теоретические основы химической кинетики, макрокинетики, теории горения и взрыва и т.д.)", i.e. (English translation) "In the Mathematical Department of the Institute (A. Ya. Povzner, A. I. Vol'pert, A. Ya. Dubovitskii) the mathematical foundations of chemical physics have been widely developed: particularly the theory of systems of differential equations, optimization techniques, advanced computational methods, imaging techniques, etc. which formed the basis of modern chemical physics (the theoretical foundations of chemical kinetics, macrokinetics, the theory of combustion and explosion, etc.)".
9. According to Ingbar (2010, p. 80).
10. According to Chuyko (2009, p. 79). See also Mikhlin (1965, pp. 185 and 207–208) and Mikhlin & Prössdorf 1986, p. 369.
11. See Mikhlin & Prössdorf 1986, p. 369 and also Prössdorf 1991, p. 108.
12. In the paper (Vol'pert 1967, pp. 246–247): see also the book (Hudjaev & Vol'pert 1985, Chapter 4, §6. "Differentiation formulas").
13. See the entry on functions of bounded variation for more details on the quantities appearing in this formula: here it is only worth to remark that a more general one, meaningful even for Lipschitz continuous functions f : ℝ^p → ℝ^s, has been proved by Luigi Ambrosio and Gianni Dal Maso in the paper (Ambrosio & Dal Maso 1990).
14. See Dal Maso, Lefloch & Murat (1995, pp. 483–484). This paper is one of several works where the results of the paper (Vol'pert 1967, pp. 246–247) are extended in order to define a particular product of distributions: the product introduced is called the "Nonconservative product".
15. See (Vol'pert 1972) and also (Hudjaev & Vol'pert 1985, pp. 607–666).

References

Biographical references

• Chuyko, Halyna I. (2009), "Functional analysis in Lviv after 1945", in Bojarski, Bogdan; Ławrynowicz, Julian; Prytula, Yaroslav G. (eds.), Lvov Mathematical School in the Period 1915–45 as Seen Today, Banach Center Publications, vol. 87, Warszawa: Institute of Mathematics – Polish Academy of Sciences, pp. 79–84, doi:10.4064/bc87-0-6, ISBN 978-83-86806-06-5, MR 2640483, Zbl 1208.01042.
• Dubovitskii, F. I. (1996), Институт химической физики. Очерки истори (in Russian), Москва: Издательство "Наука", p. 983, ISBN 5-02-010689-5. "The Institute of Chemical Physics. Historical essays" (English translation of the title) is an historical book on the Institute of Problems of Chemical Physics, written by Fedor Ivanovich Dubovitskii, one of his founders and leading directors for many years. It gives many useful details on the lives and the achievements of many scientists who worked there, including Aizik Isaakovich Vol'pert.
• Editorial staff of Focus (October 2003), "Birthday Equations" (PDF), Technion Focus: 9. A short announce of the "Partial Differential Equations and Applications" conference in celebration of Aizik I. Volpert's 80th Birthday, held in June 2003 by the Center for Mathematical Sciences, including a few biographical details. The conference participants and program can be found at the conference web site (Pinchover, Rubinstein & Shafrir 2003).
• Fomin, S. V.; Shilov, G. E., eds. (1969), Математика в СССР 1958–1967 (in Russian), vol. Том второй: Биобиблиография выпуск первый А–Л, Москва: Издательство "Наука", p. 816, MR 0250816, Zbl 0199.28501. The "Mathematics in the USSR 1958–1967" is a two–volume continuation of the opus "Mathematics in the USSR during its first forty years 1917–1957" and describes the developments of Soviet mathematics during the period 1958–1967. Precisely it is meant as a continuation of the second volume of that work and, as such, is titled "Biobibliography" (evidently an acronym of biography and bibliography). It includes new biographies (when possible, brief and complete) and bibliographies of works published by new Soviet mathematicians during that period, and updates on the work and biographies of scientist included in the former volume, alphabetically ordered with respect to author's surname.
• Ingbar, Omri, ed. (2010), "Aizik Isaakovich Volpert (1923–2006)", Outstanding Immigrant Scientists 1990–2010. Honoring Outstanding Immigrant Scientists for their Contribution to the State of Israel (in Hebrew and English), Jerusalem: Ministry of Immigrant Absorption of the State of Israel, pp. 80–81.
• Kurosh, A. G.; Vityushkov, V. I.; Boltyanskii, V. G.; Dynkin, E. B.; Shilov, G. E.; Yushkevich, A. P., eds. (1959b), Математика в СССР за сорок лет 1917–1957 (in Russian), vol. Том второй: Биобиблиография, Москва: Государственное Издательство Физико–Математическои Литературы, p. 819, MR 0115874, Zbl 0191.27501. "Mathematics in the USSR during its first forty years 1917–1957 is an opus in two volumes describing the developments of Soviet mathematics during the first forty years of its existence. This is the second volume, titled "Biobibliography" (evidently an acronym of biography and bibliography), containing a complete bibliography of works published by Soviet mathematicians during that period, alphabetically ordered with respect to author's surname and including, when possible, brief but complete biographies of the authors.
• Manelis, G. B.; Aldoshin, S. M. (2005), "Институт проблем химической физики. Пятьдесят лет на переднем крае", in Manelis, G. B. (ed.), Институт проблем химической физики, 2004. Ежегодник Том I (PDF) (in Russian), Черноголо́вка: ИПХФ РАН, pp. 5–14, ISBN 5-901675-43-6^[permanent dead link‍]. "Institute of Problems of Chemical Physics. Fifty years in the trenches" (English translation of the title) is a brief historical sketch of the institute, published in the first volume of the 2004 yearbook.
• Pinchover, Yehuda; Rubinstein, Jacob; Shafrir, Itai (11–16 June 2003), Conference on Partial Differential Equations and Applications in Celebration of Aizik I. Volpert's 80th Birthday, Haifa, Technion, retrieved 27 August 2009

Scientific references

• Ambrosio, Luigi; Dal Maso, Gianni (1990), "A General Chain Rule for Distributional Derivatives", Proceedings of the American Mathematical Society, 108 (3): 691, doi:10.1090/S0002-9939-1990-0969514-3, MR 0969514, Zbl 0685.49027.
• Dal Maso, Gianni; Lefloch, Philippe G.; Murat, François (1995), "Definition and weak stability of nonconservative products", Journal de Mathématiques Pures et Appliquées, IX Série, 74 (6): 483–548, MR 1365258, Zbl 0853.35068.
• Érdi, P.; Tóth, J. (1989), Mathematical models of chemical reactions. Theory and applications of deterministic and stochastic models, Nonlinear Science: Theory and Applications, Manchester / Princeton, N.J.: Manchester University Press / Princeton University Press, pp. xxiv+259, Bibcode:1989mmcr.book.....E, ISBN 0-7190-2208-8, MR 0981593, Zbl 0696.92027 (ISBN 0-691-08532-3 for the Princeton University Press).
• Kurosh, A. G.; Vityushkov, V. I.; Boltyanskii, V. G.; Dynkin, E. B.; Shilov, G. E.; Yushkevich, A. P., eds. (1959a), Математика в СССР за сорок лет 1917–1957 (in Russian), vol. Том пербый: Обзорные статьи, Москва: Государственное Издательство Физико–Математическои Литературы, p. 1002, MR 0115874, Zbl 0191.27501. "Mathematics in the USSR during its first forty years 1917–1957 is an opus in two volumes describing the developments of Soviet mathematics during the first forty years of its existence. This is the first volume, titled "Survey articles" and consists exactly of such kind of articles authored by Soviet experts and reviewing briefly the contributions of Soviet mathematicians to a chosen field, during the years from 1917 to 1957.
• Hudjaev, Sergei Ivanovich; Vol'pert, Aizik Isaakovich (1985), Analysis in classes of discontinuous functions and equations of mathematical physics, Mechanics: analysis, vol. 8, Dordrecht–Boston–Lancaster: Martinus Nijhoff Publishers, ISBN 90-247-3109-7, MR 0785938, Zbl 0564.46025
• Mikhlin, S.G. (1965), Multidimensional singular integrals and integral equations, International Series of Monographs in Pure and Applied Mathematics, vol. 83, Oxford-London-Edinburgh-New York-Paris-Frankfurt: Pergamon Press, pp. XII+255, MR 0185399, Zbl 0129.07701. A masterpiece in the multidimensional theory of singular integrals and singular integral equations summarizing all the results from the beginning to the year of publication, and also sketching the history of the subject.
• Prössdorf, S. (1991), "Linear Integral Equations", in Maz'ya, V. G.; Nikol'skiǐ, S. M. (eds.), Analysis IV, Encyclopaedia of Mathematical Sciences, vol. 27, Berlin–Heidelberg–New York: Springer-Verlag, pp. 1–125, ISBN 0-387-51997-1, MR 1098506, Zbl 0780.45001 (also available as ISBN 3-540-51997-1).
• Mikhlin, Solomon G.; Prössdorf, Siegfried (1986), Singular Integral Operators, Berlin-Heidelberg-New York: Springer Verlag, p. 528, ISBN 3-540-15967-3, MR 0867687, Zbl 0612.47024 (European edition ISBN 0-387-15967-3).
• Truesdell, Clifford A. III (1991) [1977], A First Course in Rational Continuum Mechanics. Volume 1: General concepts, Pure and Applied Mathematics, vol. 71 (2nd ed.), Boston – San Diego – New York – London – Sidney – Tokyo – Toronto: Academic Press, pp. xviii+391, ISBN 0-12-701300-8, MR 1162744, Zbl 0866.73001.