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Cartan–Kuranishi Prolongation Theorem
Given an exterior differential system defined on a manifold ''M'', the Cartan–Kuranishi prolongation theorem says that after a finite number of ''prolongations'' the system is either ''in involution'' (admits at least one 'large' integral manifold), or is impossible. History The theorem is named after Élie Cartan and Masatake Kuranishi. Applications This theorem is used in infinite-dimensional Lie theory In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject i .... See also * Cartan-Kähler theorem References * M. Kuranishi, ''On É. Cartan's prolongation theorem of exterior differential systems'', Amer. J. Math., vol. 79, 1957, p. 1–47 * {{DEFAULTSORT:Cartan-Kuranishi prolongation theorem Partial differential equations Theorems in analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Differential System
In the theory of differential forms, a differential ideal ''I'' is an ''algebraic ideal'' in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exterior differentiation ''d'', meaning that for any form α in ''I'', the exterior derivative ''d''α is also in ''I''. In the theory of differential algebra, a differential ideal ''I'' in a differential ring ''R'' is an ideal which is mapped to itself by each differential operator. Exterior differential systems and partial differential equations An exterior differential system consists of a smooth manifold M and a differential ideal : I\subset \Omega^*(M) . An integral manifold of an exterior differential system (M,I) consists of a submanifold N\subset M having the property that the pullback to N of all differential forms contained in I vanishes identically. One can express any partial differential equation system as an exterior different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He also made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians of the twentieth century. His son Henri Cartan was an influential mathematician working in algebraic topology. Life Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan (1837–1917) and Anne Cottaz (1841–1927). Joseph Cartan was the village blacksmith; Élie Cartan recalled that his childhood had passed under "blows of the anvil, which started every morning from dawn", and that "his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning-wheel". Élie had an elder sister J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Masatake Kuranishi
Masatake Kuranishi (倉西 æ£æ¦ ''Kuranishi Masatake''; July 19, 1924 – June 22, 2021) was a Japanese mathematician who worked on several complex variables, partial differential equations, and differential geometry. Education and career Kuranishi received in 1952 his Ph.D. from Nagoya University. He became a lecturer there in 1951, an associate professor in 1952, and a full professor in 1958. From 1955 to 1956 he was a visiting scholar at the Institute for Advanced Study in Princeton, New Jersey. From 1956 to 1961 he was a visiting professor at the University of Chicago, the Massachusetts Institute of Technology, and Princeton University. He became a professor at Columbia University in the summer of 1961. Kuranishi was an invited speaker at the International Congress of Mathematicians in 1962 at Stockholm with the talk ''On deformations of compact complex structures'' and in 1970 at Nice with the talk ''Convexity conditions related to 1/2 estimate on elliptic complexes''. He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data. Lie theory has been particularly useful in mathematical phys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
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 how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
