"Differential Galois theory"의 두 판 사이의 차이
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* basic functions : basic elementary functions | * basic functions : basic elementary functions | ||
| − | * allowed operatrions : compositions, arithmetic operations differentiation, integration | + | * allowed operatrions : compositions, arithmetic operations, differentiation, integration |
| − | * an elliptic integral is representable by quadrature | + | * examples<br> |
| + | ** an elliptic integral is representable by quadrature | ||
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<h5>Fuchsian differential equation</h5> | <h5>Fuchsian differential equation</h5> | ||
| − | * | + | * differential equation with regular singularities |
* indicial equation<br><math>x(x-1)+px+q=0</math><br> | * indicial equation<br><math>x(x-1)+px+q=0</math><br> | ||
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theorem | theorem | ||
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| + | * [[Class Field Theory]]<br> | ||
| + | * [[number fields and threefolds]]<br> | ||
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| + | * http://ko.wikipedia.org/wiki/ | ||
| + | * http://en.wikipedia.org/wiki/Differential_Galois_theory | ||
| + | * http://en.wikipedia.org/wiki/Homotopy_lifting_property | ||
| + | * http://en.wikipedia.org/wiki/covering_space | ||
| + | * http://en.wikipedia.org/wiki/Field_extension | ||
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* [http://www.jstor.org/stable/2154053 Liouvillian First Integrals of Differential Equations]<br> | * [http://www.jstor.org/stable/2154053 Liouvillian First Integrals of Differential Equations]<br> | ||
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| + | <h5>books</h5> | ||
* Group Theory and Differential Equations<br> | * Group Theory and Differential Equations<br> | ||
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* http://gigapedia.info/1/differntial+algebra | * http://gigapedia.info/1/differntial+algebra | ||
* http://gigapedia.info/1/ | * http://gigapedia.info/1/ | ||
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2010년 8월 11일 (수) 08:24 판
- differential galois theory
- Liouville
historical origin
- integration in finite terms
- quadrature of second order differential equation (Fuchsian differential equation)
differential field
solvable by quadratures
- basic functions : basic elementary functions
- allowed operatrions : compositions, arithmetic operations, differentiation, integration
- examples
- an elliptic integral is representable by quadrature
elementary extension
- using exponential and logarithm
- elementary element
Liouville extension
- we can adjoin integrals and exponentials of integrals + algbraic extension
- an element is said to be representable by a generalized quadrature
Picard-Vessiot extension
- framework for linear differential equation
- made by including solutions of DE to the base field (e.g. rational function field)
- this corresponds to the concept of the splitting fields
- we can define a Galois group for a linear differential equation.
- examples
- algebraic extension
- adjoining an integral
- adjoining the exponential of an integral
theorem
If a Picard-Vessiot extension is a Liouville extension, then the Galois group of this extension is solvable.
Fuchsian differential equation
- differential equation with regular singularities
- indicial equation
\(x(x-1)+px+q=0\)
theorem
A Fuchsian linear differential equation is solvable by quadratures if and only if the monodromy group of this equation is solvable.
solution by quadrature
- Differential Galois Theory and Non-Integrability of Hamiltonian Systems
- Integrability and non-integrability in Hamiltonian mechanics
- [1]http://www.caminos.upm.es/matematicas/morales%20ruiz/libroFSB.pdf
- http://andromeda.rutgers.edu/~liguo/DARTIII/Presentations/Khovanskii.pdf
- http://www.math.purdue.edu/~agabriel/topological_galois.pdf
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Differential_Galois_theory
- http://en.wikipedia.org/wiki/Homotopy_lifting_property
- http://en.wikipedia.org/wiki/covering_space
- http://en.wikipedia.org/wiki/Field_extension
articles
- Liouvillian First Integrals of Differential Equations
- Michael F. Singer, Transactions of the American Mathematical Society, Vol. 333, No. 2 (Oct., 1992), pp. 673-688
books
- Group Theory and Differential Equations
- Lawrence Markus, 1960
- An introduction to differential algebra
- Irving Kaplansky
- Irving Kaplansky
- algebraic theory of differential equations
- http://gigapedia.info/1/galois_theory
- http://gigapedia.info/1/differential+galois+theory
- http://gigapedia.info/1/Kolchin
- http://gigapedia.info/1/ritt
- http://gigapedia.info/1/Galois'+dream
- http://gigapedia.info/1/differntial+algebra
- http://gigapedia.info/1/