"Differential Galois theory"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
| 42번째 줄: | 42번째 줄: | ||
<h5>Picard-Vessiot extension</h5> | <h5>Picard-Vessiot extension</h5> | ||
| − | * this corresponds | + | * 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<br> | * examples<br> | ||
** algebraic extension | ** algebraic extension | ||
| 61번째 줄: | 64번째 줄: | ||
* regular singularity | * regular singularity | ||
| − | * indicial equation<br><math>x(x-1)+px+q=0</math><br> | + | * indicial equation<br><math>x(x-1)+px+q=0</math><br> |
| + | |||
| + | |||
| + | |||
| + | theorem | ||
| + | |||
| + | A Fuchsian linear differential equation | ||
2010년 1월 22일 (금) 06:42 판
- differential galois theory
- Liouville
historical origin
- integration in finite terms
- quadrature of second order differential equation (Fuchsian differential equation)
differential field
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
- regular singularity
- indicial equation
\(x(x-1)+px+q=0\)
theorem
A Fuchsian linear differential equation
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
하위페이지
관련논문
- Liouvillian First Integrals of Differential Equations
- Michael F. Singer, Transactions of the American Mathematical Society, Vol. 333, No. 2 (Oct., 1992), pp. 673-688
표준적인 도서 및 추천도서
- 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/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=