Question: Lev Pontryagin and Leonty Magnitsky were both what?

Documents:
source: Leonty Magnitsky
text: Leonty Magnitsky Leonty Filippovich Magnitsky (Russian: Леонтий Филиппович Магницкий ), born Telyatin (Russian: Телятин ), (June 9, 1669, Ostashkov – October 19, 1739, Moscow) was a Russian mathematician and educator.

source: Lev Pontryagin
text: Lev Pontryagin Lev Semyonovich Pontryagin (Russian: Лев Семёнович Понтрягин , also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight due to a primus stove explosion when he was 14. Despite his blindness he was able to become one of the greatest mathematicians of the 20th century, partially with the help of his mother Tatyana Andreevna who read mathematical books and papers (notably those of Heinz Hopf, J. H. C. Whitehead, and Hassler Whitney) to him. He made major discoveries in a number of fields of mathematics, including algebraic topology and differential topology.

source: Andronov–Pontryagin criterion
text: Andronov–Pontryagin criterion The Andronov–Pontryagin criterion is a necessary and sufficient condition for the stability of dynamical systems in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.

source: Aleksandr Andronov
text: Aleksandr Andronov Aleksandr Aleksandrovich Andronov (Russian: Алекса́ндр Алекса́ндрович Андро́нов ; April 11 [O.S. March 29] 1901 , Moscow – October 31, 1952, Gorky) was a Soviet physicist and member of the Soviet Academy of Sciences (1946). He worked extensively on the theory of stability of dynamical systems, introducing (together with Lev Pontryagin) the notion of structural stability. In that context, he also contributed to the mathematical theory of self-oscillation (a term that he coined) by establishing a link between the generation of oscillations and the theory of Lyapunov stability. He developed the comprehensive theory of self-oscillations by linking it with the qualitative theory of differential equations, topology, and with the general theory of stability of motion. The crater Andronov on the Moon is named after him.

source: Tannaka–Krein duality
text: Tannaka–Krein duality In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named for two men, the Soviet mathematician Mark Grigorievich Krein, and the Japanese Tadao Tannaka. In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category Π("G") with some additional structures, formed by the finite-dimensional representations of "G".

source: Pontryagin's maximum principle
text: Pontryagin's maximum principle Pontryagin's maximum (or minimum) principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students. It has as a special case the Euler–Lagrange equation of the calculus of variations.

source: Alexander duality
text: Alexander duality In mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by P. S. Alexandrov and Lev Pontryagin. It applies to the homology theory properties of the complement of a subspace "X" in Euclidean space, a sphere, or other manifold. It is generalized by Spanier-Whitehead duality.

source: Hamiltonian (control theory)
text: Hamiltonian (control theory) The Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. It was inspired by, but is distinct from, the Hamiltonian of classical mechanics. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to minimize the Hamiltonian. For details see Pontryagin's maximum principle.

source: Pontryagin class
text: Pontryagin class In mathematics, the Pontryagin classes, named for Lev Pontryagin, are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundles.

source: Thom space
text: Thom space In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.

source: __pad__
text: __no_content__

source: __pad__
text: __no_content__

source: __pad__
text: __no_content__

source: __pad__
text: __no_content__

source: __pad__
text: __no_content__

Answer: mathematician