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Aizik Isaakovich Vol'pert (russian: Айзик Исаакович Вольперт) (5 June 1923 – January 2006) (the family name is also transliterated as
Volpert Volpert (or Vol'pert) is a surname. Notable people with the surname include: * Aizik Isaakovich Vol'pert (1923—2006), a Soviet and Israeli mathematician. * () (1908—1988), a Soviet engineer and one of the inventors of the Smith chart. * Lari ...
or WolpertSee .) was a
Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen national ...
and
Israel Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated ...
i
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
and
chemical engineer In the field of engineering, a chemical engineer is a professional, equipped with the knowledge of chemical engineering, who works principally in the chemical industry to convert basic raw materials into a variety of products and deals with the ...
working in
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s, functions of bounded variation and
chemical kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in ...
.


Life and academic career

Vol'pert graduated from
Lviv University The University of Lviv ( uk, Львівський університет, Lvivskyi universytet; pl, Uniwersytet Lwowski; german: Universität Lemberg, briefly known as the ''Theresianum'' in the early 19th century), presently the Ivan Franko Na ...
in 1951, earning the
candidate of science Candidate of Sciences (russian: кандидат наук, translit=kandidat nauk) is the first of two doctoral level scientific degrees in Russia and the Commonwealth of Independent States. It is formally classified as UNESCO's ISCED level 8, " ...
degree and the
docent The title of docent is conferred by some European universities to denote a specific academic appointment within a set structure of academic ranks at or below the full professor rank, similar to a British readership, a French " ''maître de con ...
title respectively in 1954 and 1956 from the same university: from 1951 on he worked at the Lviv Industrial Forestry Institute. In 1961 he became
senior research fellow A research fellow is an academic research position at a university or a similar research institution, usually for academic staff or faculty members. A research fellow may act either as an independent investigator or under the supervision of a pr ...
while 1962 he earned the "
doktor nauk Doctor of Sciences ( rus, доктор наук, p=ˈdoktər nɐˈuk, abbreviated д-р наук or д. н.; uk, доктор наук; bg, доктор на науките; be, доктар навук) is a higher doctoral degree in the Russi ...
" degree from
Moscow State University M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious ...
. In the 1970s–1980s A. I. Volpert became one of the leaders of the Russian Mathematical Chemistry scientific community. He finally joined Technion’s Faculty of Mathematics in 1993, doing his
Aliyah Aliyah (, ; he, עֲלִיָּה ''ʿălīyyā'', ) is the immigration of Jews from the diaspora to, historically, the geographical Land of Israel, which is in the modern era chiefly represented by the State of Israel. Traditionally descri ...
in 1994.


Work


Index theory and elliptic boundary problems

Vol'pert developed an effective
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
for calculating the index of an elliptic problem before the Atiyah-Singer index theorem appeared: He was also the first to show that the index of a singular matrix operator can be different from zero.


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 equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s. Precisely, given a
continuously differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in i ...
and a function of
bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
with and , he proves that is again a function of bounded variation and the following
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
formula holds: :\frac=\sum_^p\frac\frac \qquad\forall i=1,\ldots,n where is the already cited functional superposition of and . By using his results, it is easy to prove that functions of bounded variation form an
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
of discontinuous functions: in particular, using his calculus for , it is possible to define the product of the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
and the Dirac distribution in one variable.


Chemical kinetics

His work on chemical kinetics and
chemical engineering Chemical engineering is an engineering field which deals with the study of operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials in ...
led him to define and study differential equations on graphs.See and also .


Selected publications

*. One of the best books about ''BV''-functions and their application to problems of
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, particularly
chemical kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in ...
. *. 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 equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s: it was also translated as . *, translated in English as . *. *. *. *. *, translated in English as . *. *. *. *.


See also

* Atiyah-Singer index theorem *
Bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
* Caccioppoli set *
Differential equation on a graph Differential may refer to: Mathematics * Differential (mathematics) comprises multiple related meanings of the word, both in calculus and differential geometry, such as an infinitesimal change in the value of a function * Differential algebra ...


Notes


References


Biographical references

*. *. "''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. *. 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 . *. The "''Mathematics in the
USSR The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nation ...
1958–1967''" is a two–volume continuation of the opus "''Mathematics in the
USSR The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nation ...
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 An acronym is a word or name formed from the initial components of a longer name or phrase. Acronyms are usually formed from the initial letters of words, as in '' NATO'' (''North Atlantic Treaty Organization''), but sometimes use syllables, a ...
of
biography A biography, or simply bio, is a detailed description of a person's life. It involves more than just the basic facts like education, work, relationships, and death; it portrays a person's experience of these life events. Unlike a profile or c ...
and
bibliography Bibliography (from and ), as a discipline, is traditionally the academic study of books as physical, cultural objects; in this sense, it is also known as bibliology (from ). English author and bibliographer John Carter describes ''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. *. *. "''Mathematics in the
USSR The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nation ...
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 An acronym is a word or name formed from the initial components of a longer name or phrase. Acronyms are usually formed from the initial letters of words, as in '' NATO'' (''North Atlantic Treaty Organization''), but sometimes use syllables, a ...
of
biography A biography, or simply bio, is a detailed description of a person's life. It involves more than just the basic facts like education, work, relationships, and death; it portrays a person's experience of these life events. Unlike a profile or c ...
and
bibliography Bibliography (from and ), as a discipline, is traditionally the academic study of books as physical, cultural objects; in this sense, it is also known as bibliology (from ). English author and bibliographer John Carter describes ''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. *. "''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 A yearbook, also known as an annual, is a type of a book published annually. One use is to record, highlight, and commemorate the past year of a school. The term also refers to a book of statistics or facts published annually. A yearbook often ...
. *


Scientific references

*. *. * ( for the
Princeton University Press Princeton University Press is an independent Academic publishing, publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, ...
). *. "''Mathematics in the
USSR The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nation ...
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. * *. A masterpiece in the multidimensional theory of
singular integral In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, w ...
s and
singular integral equation In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n ...
s summarizing all the results from the beginning to the year of publication, and also sketching the history of the subject. * (also available as ). * (European edition ). *. {{DEFAULTSORT:Volpert, Aizik Isaakovich 20th-century Israeli mathematicians Mathematical analysts Soviet chemical engineers Soviet mathematicians Russian Jews Jewish scientists 1923 births 2006 deaths Russian emigrants to Israel Israeli chemical engineers