View source for Aizik Volpert

{{Short description|Soviet and Israeli mathematician and chemical engineer}}
{{MOS|article|[[MOS:BBB]] - ℝ|date=July 2023}}
{{Infobox scientist
| name              = Aizik Isaakovich Vol'pert
| image             = <!--(filename only)-->
| image_size        = 
| alt               = 
| caption           = 
| birth_date        = {{Birth date|1923|06|05|df=y}}<ref name="biobib">See {{harvtxt|Kurosh|Vityushkov|Boltyanskii|Dynkin|1959b|p=145}}.</ref><ref name="biobib2">See {{harvtxt|Fomin|Shilov|1969|p=265}}.</ref>
| birth_place       = [[Kharkov]]
| death_date        = {{Death-date and age|January 2006|5 June 1923}}
| death_place       = [[Haifa]]
| nationality       = 
| fields            = 
| workplaces        = {{plainlist|
*[[Lviv University]]
*[[Ukrainian National Forestry University|Lviv Industrial Forestry Institute]]
*[[Technion]]}}
| alma_mater        = {{plainlist|
*[[Lviv University]]
*[[Moscow University]]<ref name="tecbio">According to the few information given by the {{harvtxt|Editorial staff of Focus|2003|p=9}}.</ref>}}
| thesis_title      = 
| thesis_url        = 
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| academic_advisors = 
| doctoral_students = 
| notable_students  = 
| known_for         = {{plainlist|
*[[Bounded variation|Functions of bounded variation]]
*[[index theory]]
*[[partial differential equation]]s
*[[parabolic partial differential equation]]s<br>[[mathematical chemistry]]}}
| influences        = 
| influenced        = 
| awards            = 
| signature         = <!--(filename only)-->
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| website           = <!-- {{URL|www.example.com}} -->
| footnotes         = 
}}
'''Aizik Isaakovich Vol'pert''' ({{langx|ru|Айзик Исаакович Вольперт}}) (5 June 1923<ref name="biobib"/><ref name="biobib2"/> – January 2006) (the family name is also transliterated as '''[[Volpert]]'''<ref>See {{harvtxt|Chuyko|2009|p=79}}.</ref> or '''Wolpert'''<ref name="Miklhin 1986 369">See {{harvnb|Mikhlin|Prössdorf|1986|p=369}}.</ref>) was a [[Soviet Union|Soviet]] and [[Israel]]i [[mathematician]] and [[chemical engineer]]<ref>His training as an engineer is clearly indicated by {{harvtxt|Truesdell|1991|p=88, footnote 1}} who, referring to the book {{Harv|Hudjaev|Vol'pert|1985}}, writes exactly:-"''Be it noted that this clear, excellent, and compact book is written by and for engineers''".</ref> working in [[partial differential equation]]s, [[Bounded variation|functions of bounded variation]] and [[chemical kinetics]].

== Life and academic career==
[[File:VolpertBykovGorbanYablonskyIvanova1980.png|thumb|400px|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, [[Alexander Nikolaevich Gorban|A.N. Gorban]], [[Grigoriy Yablonsky|G.S. Yablonsky]], A.N.Ivanova.]]
Vol'pert graduated from [[Lviv University]] in 1951, earning the [[candidate of science]] degree and the [[Docent#Central and Eastern Europe|docent]] title respectively in 1954 and 1956 from the same university:<ref name="biobib"/> from 1951 on he worked at the [[Ukrainian National Forestry University|Lviv Industrial Forestry Institute]].<ref name="biobib"/> In 1961 he became [[Research fellow#Russian Federation|senior research fellow]]<ref>Precisely he became "''старший научный сотрудник''", abbreviated as "''ст. науч. сотр.''", according to {{harvtxt|Fomin|Shilov|1969|p=265}}.</ref> while 1962 he earned the "[[doktor nauk]]"<ref name="biobib2"/> degree from [[Moscow State University]]. In the 1970s–1980s A. I. Volpert became one of the leaders of the Russian Mathematical Chemistry scientific community.<ref>{{harvtxt|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 ([[Aleksandr Yakovlevich Povzner|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.)''".</ref> He finally joined [[Technion]]’s Faculty of Mathematics in 1993,<ref name="tecbio"/> doing his [[Aliyah]] in 1994.<ref>According to {{harvtxt|Ingbar|2010|p=80}}.</ref>

== 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:<ref>According to {{harvtxt|Chuyko|2009|p=79}}. See also {{harvtxt|Mikhlin|1965|pp=185 and 207–208}} and {{harvnb|Mikhlin|Prössdorf|1986|p=369}}.</ref> He was also the first to show that the index of a singular matrix operator can be different from zero.<ref>See {{harvnb|Mikhlin|Prössdorf|1986|p=369}} and also {{harvnb|Prössdorf|1991|p=108}}.</ref>

===Functions of bounded variation===
He was one of the leading contributors to the theory of [[Bounded variation|''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 equation]]s.<ref>In the paper {{Harv|Vol'pert|1967|pp=246–247}}: see also the book {{Harv|Hudjaev|Vol'pert|1985|loc=Chapter 4, §6. "Differentiation formulas"}}.</ref> Precisely, given a [[continuously differentiable function]] {{math|''f''&nbsp;:&nbsp;ℝ<sup>''p''</sup>&nbsp;→&nbsp;ℝ}} and a function of [[bounded variation]] {{math|'''''u'''''('''''x''''')&nbsp;{{=}}&nbsp;(''u''<sub>1</sub>('''''x'''''),...,''u<sub>p</sub>''('''''x'''''))}} with {{math|'''''x''''' ∈ ℝ<sup>''n''</sup>}} and {{math|''n'' ≥ 1}}, he proves that {{math|''f''∘'''''u'''''('''''x''''')&nbsp;{{=}}&nbsp;''f''('''''u'''''('''''x'''''))}} is again a function of bounded variation and the following [[chain rule]] [[formula]] holds:<ref>See the entry on [[Bounded variation|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 continuity|Lipschitz continuous functions]] {{math|'''''f''''' : ℝ<sup>''p''</sup>&nbsp;→&nbsp;ℝ<sup>''s''</sup>}}, has been proved by [[Luigi Ambrosio]] and [[Gianni Dal Maso]] in the paper {{Harv|Ambrosio|Dal Maso|1990}}.</ref>

:<math>\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</math>

where {{math|''{{overset|'''–'''|f}}''('''''u'''''('''''x'''''))}} is the already cited functional superposition of  {{math|''f''}} and {{math|'''''u'''''}}. By using his results, it is easy to prove that functions of bounded variation form an [[Algebra (ring theory)|algebra]] of [[discontinuous function]]s: in particular, using his calculus for {{math|''n''&nbsp;{{=}}&nbsp;1}}, it is possible to define the product {{math|H&nbsp;⋅&nbsp;''δ''}} of the [[Heaviside step function]] {{math|H(''x'')}} and the [[Dirac distribution]] {{math|''δ''(''x'')}}  in one [[Variable (mathematics)|variable]].<ref>See {{harvtxt|Dal Maso|Lefloch|Murat|1995|pp=483–484}}. This paper is one of several works where the results of the paper {{Harv|Vol'pert|1967|pp=246–247}} are extended in order to define a particular [[Generalized function#Multiplication of distributions|product of distributions]]: the product introduced is called the "''[[Nonconservative product]]''".</ref>

===Chemical kinetics===

His work on chemical kinetics and [[chemical engineering]] led him to define and study [[differential equation on a graph|differential equations on graphs]].<ref>See {{Harv|Vol'pert|1972}} and also {{Harv|Hudjaev|Vol'pert|1985|pp=607–666}}.</ref>

==Selected publications==
*{{Citation
|last1 = Hudjaev
|first1 = Sergei Ivanovich
|last2 = Vol'pert
|first2 = Aizik Isaakovich
|title = Analysis in classes of discontinuous functions and equations of mathematical physics
|place =  Dordrecht–Boston–Lancaster
|publisher = [[Martinus Nijhoff Publishers]]
|year = 1985
|series = Mechanics: analysis 
|volume = 8
|pages = xviii+678 
|url = https://books.google.com/books?id=lAN0b0-1LIYC&q=%22Analysis+in+classes+of+discontinuous+functions%22
|mr = 0785938
|zbl= 0564.46025
|isbn = 90-247-3109-7
|ref = none
}}. One of the best books about ''BV''-functions and their application to problems of [[mathematical physics]], particularly [[chemical kinetics]].
*{{Citation
|last = Vol'pert
|first = Aizik Isaakovich
|script-title = ru:Пространства BV и квазилинейные уравнени
|journal = [[Matematicheskii Sbornik]]
|series = (N.S.)
|volume = 73(115)
|language = ru
|issue = 2
|pages = 255–302
|year = 1967
|url = http://mi.mathnet.ru/eng/msb/v115/i2/p255
|mr = 216338 
|zbl = 0168.07402
}}. A seminal paper where [[Caccioppoli set]]s and ''BV'' functions are thoroughly studied and the concept of [[functional superposition]] is introduced and applied to the theory of [[partial differential equation]]s: it was also translated as {{Citation
|title = Spaces BV and quasi-linear equations
|journal = [[Sbornik: Mathematics|Mathematics of the USSR-Sbornik]]
|volume = 2
|issue = 2
|pages = 225–267
|year = 1967
|doi = 10.1070/SM1967v002n02ABEH002340
|mr = 216338
|zbl = 0168.07402
|last1 = Vol'Pert
|first1 = A. I.
|bibcode = 1967SbMat...2..225V
|hdl = 10338.dmlcz/102500
|hdl-access = free
}}.
*{{Citation
|last = Vol'pert
|first = Aizik Isaakovich
|script-title = ru:Дифференциальные уравнения на графах
|journal = [[Matematicheskii Sbornik]]
|series= (N.S.)
|language = ru
|volume = 88(130)
|issue = 4(8)
|pages = 578–588
|year = 1972
|url = http://mi.mathnet.ru/eng/msb/v130/i4/p578
|mr = 316796
|zbl = 0242.35015
}}, translated in English as {{Citation
|title = Differential equations on graphs
|journal = [[Sbornik: Mathematics|Mathematics of the USSR-Sbornik]]
|volume = 17
|issue = 4
|pages = 571–582
|year = 1972
|doi = 10.1070/SM1972v017n04ABEH001603
|zbl = 0255.35013
|last1 = Vol'Pert
|first1 = A. I.
|bibcode = 1972SbMat..17..571V
|url = http://www.mmnp-journal.org/10.1051/mmnp/201510502/pdf
}}.
*{{Citation
|last1 = Vasiliev
|first1 = V. M.
|last2 = Volpert
|first2 = A. I.
|last3 = Hudiaev
|first3 = S. I.
|title = On the method of quasi-stationary concentrations for chemical kinetics equations
|journal =  [[Zhurnal Vychislitel'noi Matematiki I Matematicheskoi Fiziki|Журнал вычислительной математики и математической физики]]
|volume = 13
|issue = 3
|pages = 683–697
|year = 1973
|language = ru
}}.
*{{Citation
|last = Vol'pert
|first = A. I.
|title = Qualitative methods of investigation of equations of chemical kinetics
|journal = Preprint
|place = Institute of Chemical Physics, Chernogolovka
|year = 1976
|language = ru
}}.
*{{Citation
|last1 = Vol'pert
|first1 = V. A.
|last2 =  Vol'pert
|first2 = A. I.
|last3 = Merzhanov
|first3 = A. G.
|title = Application of the theory of bifurcations in study of the spinning combustion waves
|journal = [[Doklady Akademii Nauk SSSR]]
|volume = 262
|issue = 3
|pages = 642–645
|year = 1982
|language = ru
}}.
*{{Citation
|last1 = Vol'pert
|first1 = V. A.
|last2 =  Vol'pert
|first2 = A. I.
|last3 = Merzhanov
|first3 = A. G.
|title = Analysis of nonunidimensional combustion modes by bifurcation theory methods
|journal = [[Doklady Akademii Nauk SSSR]]
|volume = 263
|issue = 4
|pages = 918–921
|year = 1982b
|language = ru
}}.
*{{Citation
|last1 = Vol'pert
|first1 = V. A.
|last2 =  Vol'pert
|first2 = A. I.
|last3 = Merzhanov
|first3 = A. G.
|title = Application of the theory of bifurcations to the study of unsteady regimes of combustion
|journal = Fizika Goreniya i Vzryva
|volume = 19
|pages = 69–72
|year = 1983
|language = ru
}}, translated in English as {{Citation
|title = Application of the theory of bifurcations to the investigation of nonstationary combustion regimes
|journal = Combustion, Explosion, and Shock Waves
|volume = 19
|issue = 4
|pages = 435–438
|year = 1983
|doi = 10.1007/BF00783642
|last1 = Vol'Pert
|first1 = V. A.
|last2 = Vol'Pert
|first2 = A. I.
|last3 = Merzhanov
|first3 = A. G.
|bibcode = 1983CESW...19..435V
|s2cid = 97950149
}}.
*{{Citation
|last1 = Vol'pert
|first1 = V. A.
|last2 =  Vol'pert
|first2 = A. I.
|contribution = Existence and stability of traveling waves in chemical kinetics
|title = Dynamics of Chemical and Biological Systems
|place = Novosibirsk
|publisher = [[Nauka (publisher)|Nauka]]
|year = 1989
|pages = 56–131
|language = ru
}}.
*{{Citation
|last1 = Vol'pert
|first1 = Aizik I.
|last2 = Vol'pert
|first2 = Vitaly A. 
|last3 = Vol'pert
|first3 = Vladimir A. 
|title = Traveling Wave Solutions of Parabolic Systems
|place = [[Providence, R.I.]]
|publisher = [[American Mathematical Society]]
|year = 1994
|series = Translations of Mathematical Monographs
|pages = xii+448
|volume = 140
|mr = 1297766
|zbl = 1001.35060
|isbn = 0-8218-3393-6
}}.
*{{Citation
|first = A. I.
|last = Vol'pert
|editor-last = Oleinik
|editor-first = O. A.
|editor-link = Olga Arsenievna Oleinik
|contribution = Propagation of Waves Described by Nonlinear Parabolic Equations (a commentary on article 6)
|contribution-url = https://books.google.com/books?id=GsJALnwAvm8C&q=Ivan+Georgievich+Petrowsky++Alexandrov+Oleinik&pg=PA1
|title = I. G. Petrovsky Selected works. Part II: Differential equations and probability theory
|place = [[Amsterdam]]
|publisher = [[Taylor & Francis|Gordon and Breach Publishers]]
|pages = 364–399
|year = 1996
|series = Classics of Soviet Mathematics
|volume = 5 (part 2)
|url = https://books.google.com/books?id=FdCuk2YumowC
|isbn = 2-88124-979-5
|mr = 1677648
|zbl = 0948.01043}}.
*{{Citation
|last1 = Vol'pert
|first1 = V. A.
|last2 =  Vol'pert
|first2 = A. I.
|title = Convective instability of reaction fronts: linear stability analysis
|journal = [[European Journal of Applied Mathematics]]
|volume = 9 
|issue = 5
|pages = 507–525
|year = 1998
|doi =10.1017/S095679259800357X
|mr = 1662311
|zbl = 0918.76027
|s2cid = 121533943
}}.

== See also ==
*[[Atiyah-Singer index theorem]]
*[[Bounded variation]]
*[[Caccioppoli set]]
*[[Differential equation on a graph]]

==Notes==
{{Reflist|29em}}

==References==
{{refbegin}}

===Biographical references===
*{{Citation
|first = Halyna I.
|last =  Chuyko
|editor-last = Bojarski
|editor-first = Bogdan 
|editor2-last = Ławrynowicz 
|editor2-first = Julian
|editor3-last = Prytula 
|editor3-first = Yaroslav G. 
|contribution = Functional analysis in Lviv after 1945
|contribution-url = http://journals.impan.pl/cgi-bin/bc/pdf?bc87-0-06
|title = Lvov Mathematical School in the Period 1915–45 as Seen Today
|series = Banach Center Publications
|volume = 87
|year = 2009
|pages = 79–84
|place = Warszawa
|publisher = Institute of Mathematics – [[Polish Academy of Sciences]]
|doi = 10.4064/bc87-0-6
|isbn = 978-83-86806-06-5
|mr = 2640483
|zbl = 1208.01042
|chapter-url-access=subscription
|doi-access = free
}}.
*{{Citation
|last = Dubovitskii
|first = F. I.
|author-link = Fedor Ivanovich Dubovitskii
|script-title=ru:Институт химической физики. Очерки истори
|place = [[Москва]]
|publisher = Издательство "[[Nauka (publisher)|Наука]]"
|year = 1996
|pages = 983
|language = ru
|url = http://stai2005.narod.ru/iz_dubovickogo.html
|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.
*{{Citation
|last = Editorial staff of Focus
|title = Birthday Equations
|journal = [[Technion Focus]]
|date=October 2003
|page = 9
|url = http://www.focus.technion.ac.il/previousIssues/FocusOct03.pdf
}}. 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 [[Technion Center for Mathematical Sciences|Center for Mathematical Sciences]], including a few biographical details. The conference participants and program can be found at the conference web site {{harv|Pinchover|Rubinstein|Shafrir|2003}}.
*{{Citation
|editor-last = Fomin
|editor-first = S. V.
|editor-link = Sergei Fomin
|editor2-last = Shilov
|editor2-first = G. E.
|editor2-link = Georgiy Shilov
|script-title=ru:Математика в СССР 1958–1967
|place = [[Москва]]
|publisher = Издательство "[[Nauka (publisher)|Наука]]"
|volume = Том второй: Биобиблиография выпуск первый А–Л
|year = 1969
|pages = 816
|language = ru
|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.
*{{Citation
|editor-last = Ingbar
|editor-first = Omri  
|contribution = Aizik Isaakovich Volpert (1923–2006)
|title = Outstanding Immigrant Scientists 1990–2010. Honoring Outstanding Immigrant Scientists for their Contribution to the State of Israel
|year = 2010
|pages = 80–81
|place = Jerusalem
|language=he, en
|publisher = [[Ministry of Immigrant Absorption|Ministry of Immigrant Absorption of the State of Israel]]
}}.
*{{Citation
|editor-last = Kurosh
|editor-first = A. G.
|editor-link = Aleksandr Gennadievich Kurosh
|editor2-last = Vityushkov
|editor2-first = V. I.
|editor3-last = Boltyanskii
|editor3-first = V. G. 
|editor3-link = Vladimir Grigorevich Boltyansky
|editor4-last = Dynkin
|editor4-first = E. B. 
|editor4-link = Eugene Borisovich Dynkin
|editor5-last = Shilov
|editor5-first = G. E.
|editor5-link = Georgiy Shilov
|editor6-last = Yushkevich
|editor6-first = A. P.
|script-title=ru:Математика в СССР за сорок лет 1917–1957
|place = [[Москва]]
|publisher = Государственное Издательство Физико–Математическои Литературы
|volume = Том второй: Биобиблиография
|year = 1959b
|pages = 819
|language = ru
|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.
*{{Citation
 |first1=G. B. 
 |last1=Manelis 
 |first2=S. M. 
 |last2=Aldoshin 
 |editor-last=Manelis 
 |editor-first=G. B. 
 |contribution=Институт проблем химической физики. Пятьдесят лет на переднем крае 
 |title=Институт проблем химической физики, 2004. Ежегодник Том I 
 |year=2005 
 |pages=5–14 
 |place=[[Chernogolovka|Черноголо́вка]] 
 |publisher=[[Institute of Problems of Chemical Physics|ИПХФ РАН]] 
 |language=ru 
 |url=http://ftp.icp.ac.ru/achievements/egegodnik/pdf/IPCP_RAS_2004.pdf
 |isbn=5-901675-43-6 
}}{{dead link|date=June 2017 |bot=InternetArchiveBot |fix-attempted=yes }}. "''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]].
*{{Citation
|last1 = Pinchover 
|first1 = Yehuda
|last2 = Rubinstein 
|first2 = Jacob
|last3 = Shafrir  
|first3 = Itai  
|title = Conference on Partial Differential Equations and Applications in Celebration of Aizik I. Volpert's 80th Birthday
|date = 11–16 June 2003
|place = Haifa, [[Technion]]
|url = https://www.math.technion.ac.il/~shafrir/volpert.html
|access-date = 27 August 2009}}

===Scientific references===
*{{Citation
|last1 = Ambrosio
|first1 = Luigi
|author-link = Luigi Ambrosio
|last2 = Dal Maso
|first2 = Gianni
|title = A General Chain Rule for Distributional Derivatives
|journal = [[Proceedings of the American Mathematical Society]]
|volume = 108
|issue = 3
|pages = 691
|year = 1990
|doi = 10.1090/S0002-9939-1990-0969514-3
|mr = 969514 
|zbl = 0685.49027
|doi-access = free
}}.
*{{Citation
|last1 = Dal Maso
|first1 =  Gianni
|last2 = Lefloch
|first2 = Philippe G.
|last3 = Murat
|first3 = François
|title = Definition and weak stability of nonconservative products  
|journal = [[Journal de Mathématiques Pures et Appliquées]]
|series = IX Série
|volume = 74
|issue = 6
|pages = 483–548
|year = 1995
|mr = 1365258
|zbl = 0853.35068
}}.
*{{Citation
|last1 = Érdi
|first1 = P.
|last2 = Tóth
|first2 =  J.
|title = Mathematical models of chemical reactions. Theory and applications of deterministic and stochastic models
|place = [[Manchester]] / [[Princeton, N.J.]]
|publisher = [[Manchester University Press]] / [[Princeton University Press]]
|series = Nonlinear Science: Theory and Applications
|year = 1989
|pages = 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]]).
*{{Citation
|editor-last = Kurosh
|editor-first = A. G.
|editor-link = Aleksandr Gennadievich Kurosh
|editor2-last = Vityushkov
|editor2-first = V. I.
|editor3-last = Boltyanskii
|editor3-first = V. G. 
|editor3-link = Vladimir Grigorevich Boltyansky
|editor4-last = Dynkin
|editor4-first = E. B. 
|editor4-link = Eugene Borisovich Dynkin
|editor5-last = Shilov
|editor5-first = G. E.
|editor5-link = Georgiy Shilov
|editor6-last = Yushkevich
|editor6-first = A. P.
|script-title=ru:Математика в СССР за сорок лет 1917–1957
|place = [[Москва]]
|publisher = Государственное Издательство Физико–Математическои Литературы
|volume = Том пербый: Обзорные статьи
|year = 1959a
|pages = 1002
|language = ru
|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.
*{{Citation
| last1 = Hudjaev
| first1 = Sergei Ivanovich 
| last2 = Vol'pert
| first2 = Aizik Isaakovich
| author2-link = Aizik Isaakovich Vol'pert
| title = Analysis in classes of discontinuous functions and equations of mathematical physics
| place = Dordrecht–Boston–Lancaster
| publisher = Martinus Nijhoff Publishers
| year = 1985
| series = Mechanics: analysis
| volume = 8
| url = https://books.google.com/books?id=lAN0b0-1LIYC&q=%22Analysis+in+classes+of+discontinuous+functions%22
| mr = 785938 
| isbn = 90-247-3109-7
| zbl = 0564.46025
}}
*{{Citation
|last = Mikhlin
|first = S.G.
|author-link = Solomon Mikhlin
|title = Multidimensional singular integrals and integral equations
|place = [[Oxford]]-London-[[Edinburgh]]-New York-Paris-[[Frankfurt]]
|publisher = [[Pergamon Press]]
|year = 1965
|series = International Series of Monographs in Pure and Applied Mathematics
|volume = 83
|mr = 0185399
|zbl = 0129.07701
|pages = XII+255
}}. A masterpiece in the [[dimension|multidimensional]] theory of [[singular integral]]s and [[singular integral equation]]s summarizing all the results from the beginning to the year of publication, and also sketching the history of the subject.
*{{Citation
|last = Prössdorf
|first = S. 
|author-link =Siegfried Prössdorf
|editor-last = Maz'ya
|editor-first = V. G.
|editor2-last = Nikol'skiǐ
|editor2-first = S. M.
|editor2-link = Sergey Nikolsky
|contribution = Linear Integral Equations
|title = Analysis IV 
|publisher = [[Springer-Verlag]] 
|place = Berlin–Heidelberg–New York
|year = 1991
|volume = 27
|series = Encyclopaedia of Mathematical Sciences
|pages = 1–125
|isbn = 0-387-51997-1
|mr =1098506
|zbl = 0780.45001
}} (also available as {{ISBN|3-540-51997-1}}).
*{{Citation
|last1 = Mikhlin
|first1 = Solomon G.
|author-link = Solomon Mikhlin
|last2 = Prössdorf
|first2 = Siegfried 
|title = Singular Integral Operators
|place = Berlin-[[Heidelberg]]-New York
|publisher = [[Springer Verlag]]
|year = 1986 
|pages = 528
|url = https://books.google.com/books?id=eaMmy99UTHgC
|mr = 0867687 
|zbl = 0612.47024
|isbn = 3-540-15967-3}} (European edition {{ISBN|0-387-15967-3}}).
*{{Citation
|last = Truesdell
|first = Clifford A. III
|author-link = Clifford A. Truesdell
|title = A First Course in Rational Continuum Mechanics. Volume 1: General concepts
|place = Boston – San Diego – New York – London – Sidney – Tokyo – Toronto
|publisher = [[Academic Press]]
|series = Pure and Applied Mathematics
|volume = 71
|orig-year = 1977
|year = 1991
|edition = 2nd
|pages = xviii+391
|url = https://books.google.com/books?id=l5J3oQ6V5RsC
|isbn = 0-12-701300-8
|mr =  1162744 
|zbl = 0866.73001
}}.
{{refend}}

{{authority control}}

{{DEFAULTSORT:Volpert, Aizik Isaakovich}}
[[Category:20th-century Israeli mathematicians]]
[[Category:Mathematical analysts]]
[[Category:Soviet chemical engineers]]
[[Category:Soviet mathematicians]]
[[Category:Russian Jews]]
[[Category:Jewish scientists]]
[[Category:1923 births]]
[[Category:2006 deaths]]
[[Category:Russian emigrants to Israel]]
[[Category:Israeli chemical engineers]]