Aizik Volpert

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

Life and academic career

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: from 1951 on he worked at the Lviv Industrial Forestry Institute. In 1961 he became senior research fellow while 1962 he earned the "doktor nauk" degree from Moscow State University. 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 in 1994.

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: 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 equations. Precisely, given a continuously differentiable function

f

:

R

p

→

R

{\displaystyle f:\mathbb {R} ^{p}\to \mathbb {R} }

and a function of bounded variation

u

(

x

)

=

(

u

1

(

x

)

,

…

,

u

p

(

x

)

)

{\displaystyle {\boldsymbol {u}}({\boldsymbol {x}})=(u_{1}({\boldsymbol {x}}),\ldots ,u_{p}({\boldsymbol {x}}))}

with

x

∈

R

n

{\displaystyle {\boldsymbol {x}}\in {\mathbb {R}}^{n}}

and

n

≥

1

{\displaystyle n\geq 1}

, he proves that

f

∘

u

(

x

)

=

f

(

u

(

x

)

)

{\displaystyle f\circ {\boldsymbol {u}}({\boldsymbol {x}})=f({\boldsymbol {u}}({\boldsymbol {x}}))}

is again a function of bounded variation and the following chain rule formula holds:

∂

f

(

u

(

x

)

)

∂

x

i

=

∑

k

=

1

p

∂

f

¯

(

u

(

x

)

)

∂

u

k

∂

u

k

(

x

)

∂

x

i

∀

i

=

1

,

…

,

n

{\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

)

)

{\displaystyle {\bar {f}}({\boldsymbol {u}}({\boldsymbol {x}}))}

is the already cited functional superposition of

f

{\displaystyle f}

and

u

{\displaystyle {\boldsymbol {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

{\displaystyle n=1}

, it is possible to define the product

H

⋅

δ

{\displaystyle H\cdot \delta }

of the Heaviside step function

H

(

x

)

{\displaystyle H(x)}

and the Dirac distribution

δ

(

x

)

{\displaystyle \delta (x)}

in one variable.

Chemical kinetics

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