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]
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 functionf : ℝp → ℝ and a function of bounded variationu(x) = (u1(x),...,up(x)) with x ∈ ℝn and n ≥ 1, he proves that f∘u(x) = f(u(x)) is again a function of bounded variation and the following chain ruleformula holds:[13]
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 functionH(x) and the Dirac distributionδ(x) in one variable.[14]
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, doi:10.1007/BF00783642, S2CID97950149.
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.
^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".
^Precisely he became "старший научный сотрудник", abbreviated as "ст. науч. сотр.", according to Fomin & Shilov (1969, p. 265).
^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.)".
^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".
Dubovitskii, F. I. (1996), Институт химической физики. Очерки истори (in Russian), Москва: Издательство "Наука", p. 983, ISBN5-02-010689-5. "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, MR0250816, Zbl0199.28501. 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, MR0115874, Zbl0191.27501. "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, ISBN5-901675-43-6[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.
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, MR0115874, Zbl0191.27501. "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.