# Aizik Volpert

(Redirected from Aizik Isaakovich Vol'pert)
Aizik Isaakovich Vol'pert
Born June 5, 1923[1][2]
Kharkov
Died January 2006 (aged 82)
Haifa
Institutions
Alma mater
Known for

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.

The Programm Committee of the Russian Conference "Mathematical Metods 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 is 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 : ℝp → ℝ and a function of bounded variation u(x) = (u1(x),...,up(x)) with x ∈ ℝn and n ≥ 1, he proves that fu(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 (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]

## Notes

1. ^ a b c d See Kurosh et al. (1959b, p. 145).
2. ^ a b c See Fomin & Shilov (1969, p. 265).
3. ^ a b 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 1986), 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 Miklhin & Prössdorf (1986, p. 369).
11. ^ See Mikhlin & Prössdorf (1986, p. 369) and also (Prössdorf, p. 108).
12. ^ In the paper (Vol'pert 1967, pp. 246–247): see also the book (Hudjaev & Vol'pert 1985, Chapter 4, §6. "Differetiation 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 : ℝp → ℝs, 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).