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Cartan–Kähler Theorem
In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I. It is named for Élie Cartan and Erich Kähler. Meaning It is not true that merely having dI contained in I is sufficient for integrability. There is a problem caused by singular solutions. The theorem computes certain constants that must satisfy an inequality in order that there be a solution. Statement Let (M,I) be a real analytic EDS. Assume that P \subseteq M is a connected, ''k''-dimensional, real analytic, regular integral manifold of ''I'' with r(P) \geq 0 (i.e., the tangent spaces T_p P are "extendable" to higher dimensional integral elements). Moreover, assume there is a real analytic submanifold R \subseteq M of codimension r(P) containing P and such that T_pR \cap H(T_pP) has dimension k+1 for all p \in P. Then there exists a (locally) unique connected, (k+1)-dimensional, real analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Ideal
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 differ ... [...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 differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrability Conditions For Differential Systems
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form ''restricts'' to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example, including Lax pairs of integrable systems. A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to find ''solutions'' to the system). Given a collection of differential 1-forms \textstyle\alpha_i, i=1,2,\dots, k on an \textstyle n-dimensional manifold M, an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its ... [...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] |
Erich Kähler
Erich Kähler (; 16 January 1906 – 31 May 2000) was a German mathematician with wide-ranging interests in geometry and mathematical physics, who laid important mathematical groundwork for algebraic geometry and for string theory. Education and life Erich Kähler was born in Leipzig, the son of a telegraph inspector Ernst Kähler. Inspired as a boy to be an explorer after reading books about Sven Hedin that his mother Elsa Götsch had given to him, the young Kähler soon focused his passion for exploration on astronomy. He is said to have written a 50-page thesis on fractional differentiation while still in high school, hoping that it would earn him a PhD. His teachers replied that he would have to attend university courses first. Kähler enrolled in the University of Leipzig in 1924. He read Galois theory, met the mathematician Emil Artin, and did research under the supervision of Leon Lichtenstein. Still fascinated by celestial mechanics, Kähler wrote a dissertation entitle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Solution
A singular solution ''ys''(''x'') of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full real line. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. In some cases, the term ''singular solution'' is used to mean a solution at which there is a failure of uniqueness to the initial value problem at every point on the curve. A singular solution in this stronger sense is often given as tangent to every solution from a family of solutions. By ''tangent'' we mean that there is a point ''x'' where ''ys''(''x'') = ''yc''(''x'') and ''y's''(''x'') = ''y'c''(''x'') where ''yc'' is a solution in a family of solutions parameterized by ''c''. This m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Manifold
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form ''restricts'' to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example, including Lax pairs of integrable systems. A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to find ''solutions'' to the system). Given a collection of differential 1-forms \textstyle\alpha_i, i=1,2,\dots, k on an \textstyle n-dimensional manifold M, an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Dieudonné
Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the ''Éléments de géométrie algébrique'' project of Alexander Grothendieck, and as a historian of mathematics, particularly in the fields of functional analysis and algebraic topology. His work on the classical groups (the book ''La Géométrie des groupes classiques'' was published in 1955), and on formal groups, introducing what now are called Dieudonné modules, had a major effect on those fields. He was born and brought up in Lille, with a formative stay in England where he was introduced to algebra. In 1924 he was admitted to the École Normale Supérieure, where André Weil was a classmate. He began working in complex analysis. In 1934 he was one of the group of ''normaliens'' convened by Weil, which would become ... [...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] |
