"Differential Galois theory"의 두 판 사이의 차이
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Liouville extension==
imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | ==introduction | + | ==introduction== |
* differential galois theory | * differential galois theory | ||
| 9번째 줄: | 9번째 줄: | ||
| − | ==historical origin | + | ==historical origin== |
* integration in finite terms | * integration in finite terms | ||
| 18번째 줄: | 18번째 줄: | ||
| − | ==solution by quadrature | + | ==solution by quadrature== |
* [http://pythagoras0.springnote.com/pages/4913609 일계 선형미분방정식]<br><math>\frac{dy}{dx}+a(x)y=b(x)</math><br><math>y(x)e^{\int a(x)\,dx}=\int b(x)e^{\int a(x)\,dx} \,dx+C</math><br> | * [http://pythagoras0.springnote.com/pages/4913609 일계 선형미분방정식]<br><math>\frac{dy}{dx}+a(x)y=b(x)</math><br><math>y(x)e^{\int a(x)\,dx}=\int b(x)e^{\int a(x)\,dx} \,dx+C</math><br> | ||
| 28번째 줄: | 28번째 줄: | ||
| − | ==differential field | + | ==differential field== |
* a pair <math>(F,\partial)</math> such that<br> | * a pair <math>(F,\partial)</math> such that<br> | ||
| 39번째 줄: | 39번째 줄: | ||
| − | ==solvable by quadratures | + | ==solvable by quadratures== |
* basic functions : basic elementary functions | * basic functions : basic elementary functions | ||
| 50번째 줄: | 50번째 줄: | ||
| − | ==elementary extension | + | ==elementary extension== |
* it is allowed to take exponentials and logarithms to make a field extension | * it is allowed to take exponentials and logarithms to make a field extension | ||
| 62번째 줄: | 62번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Liouville extension | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Liouville extension== |
* an element is said to be representable by a generalized quadrature | * an element is said to be representable by a generalized quadrature | ||
| 92번째 줄: | 92번째 줄: | ||
| − | ==Picard-Vessiot extension | + | ==Picard-Vessiot extension== |
* framework for linear differential equation | * framework for linear differential equation | ||
| 116번째 줄: | 116번째 줄: | ||
| − | ==Fuchsian differential equation | + | ==Fuchsian differential equation== |
* differential equation with regular singularities | * differential equation with regular singularities | ||
| 131번째 줄: | 131번째 줄: | ||
| − | ==solution by quadrature | + | ==solution by quadrature== |
* [http://www.caminos.upm.es/matematicas/morales%20ruiz/libroFSB.pdf Differential Galois Theory and Non-Integrability of Hamiltonian Systems] | * [http://www.caminos.upm.es/matematicas/morales%20ruiz/libroFSB.pdf Differential Galois Theory and Non-Integrability of Hamiltonian Systems] | ||
| 143번째 줄: | 143번째 줄: | ||
| − | <h5 class="r">related items | + | <h5 class="r">related items== |
* [[Class Field Theory]]<br> | * [[Class Field Theory]]<br> | ||
| 155번째 줄: | 155번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">encyclopedia | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">encyclopedia== |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 167번째 줄: | 167번째 줄: | ||
| − | <h5 class="r">articles | + | <h5 class="r">articles== |
* [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> | ||
| 178번째 줄: | 178번째 줄: | ||
| − | ==books | + | ==books== |
* Group Theory and Differential Equations<br> | * Group Theory and Differential Equations<br> | ||
2012년 10월 28일 (일) 14:26 판
introduction
- differential galois theory
- Liouville
- 2008년 12월 9일 MCF 'differential Galois theory'
historical origin
- integration in finite terms
- quadrature of second order differential equation (Fuchsian differential equation)
solution by quadrature
- 일계 선형미분방정식
\(\frac{dy}{dx}+a(x)y=b(x)\)
\(y(x)e^{\int a(x)\,dx}=\int b(x)e^{\int a(x)\,dx} \,dx+C\) - \(y''-2xy'=0\)
\(y=\int e^{x^2}\, dx\) - note that the integral of an exponential naturally shows up in expression solutions
differential field
- a pair \((F,\partial)\) such that
- \(\partial(a+b)=\partial a+\partial b\)
- \(\partial(ab)=(\partial a)b+a(\partial b)\)
- \(C_F=\ker \partial\)
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
- it is allowed to take exponentials and logarithms to make a field extension
- elementary element
- difference between Liouville extension
- exponential+ integral <=> differentiation + exponential of integral
- in elementary extension, we are not allowed to get an integrated element
Liouville extension==
- an element is said to be representable by a generalized quadrature
- we can capture these properties using the concept of Liouville extension
- to get a Liouville extension, we can adjoin
- integrals
- exponentials of integrals
- algebraic extension (generalized Liouville extension)
- from these we can include the following operations
- exponential
- logarithm
- For\(K_{i}=K_{i-1}(e_i)\) , one of the following condition holds
- \(e_i'\in K_{i-1}\), i.e. \(e_i=\int e_i'\in K_i\)
- \(e_{i}'/e_{i}\in K_{i-1}\) i.e. \((\log e_i)' \in K_{i-1}\)
- \(e_{i}\) is algebraic over \(K_{i-1}\)
- remark on exponentiation
- Let \(a,a'\in F\). Is \(b=e^a\in K\) where K is a Liouville extension?
- \(b'=a' e^a=a'b\) implies \(a'=\frac{b'}{b}\in F\).
- the exponential of the integral of a' i.e. \(e^{\int a'}=e^a+c\) must be in the Liouville extension. So \(b=e^a\in K\).
- remark on logarithm
- \(b=\log a\) is the integral of \(a'/a\in F\). So \(b\in K\)
- a few result
- K/F is a Liouville extension iff the differential Galois group K over F is solvable.
- K/F is a generalized Liouville extension iff the differential Galois group K over F has the solvable component of the identity
Picard-Vessiot extension
- framework for linear differential equation
- field extension is made by including solutions of DE to the base field (e.g. rational function field)
- consider monic differential equations over a differential field F
\(\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0\), \(a_i\in F\)
- \((E,\partial_E)\supseteq (F,\partial_F)\) is a Picard-Vessiot extension for \(\mathcal{L}\) if
- E/F is generated by n linear independent solution to \(\mathcal{L}\), i.e. adjoining basis of \(V=\mathcal{L}^{-1}(0)\) to F
- \(C_E=C_F\), \(\partial_E\mid_F=\partial_F\)
- this corresponds to the concept of the splitting fields(or Galois extensions)
- examples
- algebraic extension
- adjoining an integral
- adjoining the exponential of an integral
- we can define a Galois group for a linear differential equation
\(\operatorname{Gal}(E/F)=\{\sigma\in\operatorname{Aut}E|\partial(\sigma(n))=\sigma(\partial a), \sigma\mid _F=\operatorname{id} \}\)
- the action of an element of the Galois group is determined by its action on a basis of V
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
- integrals
- exponentials of integrals
- algebraic extension (generalized Liouville extension)
- from these we can include the following operations
- exponential
- logarithm
- from these we can include the following operations
- \(e_i'\in K_{i-1}\), i.e. \(e_i=\int e_i'\in K_i\)
- \(e_{i}'/e_{i}\in K_{i-1}\) i.e. \((\log e_i)' \in K_{i-1}\)
- \(e_{i}\) is algebraic over \(K_{i-1}\)
- Let \(a,a'\in F\). Is \(b=e^a\in K\) where K is a Liouville extension?
- \(b'=a' e^a=a'b\) implies \(a'=\frac{b'}{b}\in F\).
- the exponential of the integral of a' i.e. \(e^{\int a'}=e^a+c\) must be in the Liouville extension. So \(b=e^a\in K\).
- \(b=\log a\) is the integral of \(a'/a\in F\). So \(b\in K\)
- K/F is a Liouville extension iff the differential Galois group K over F is solvable.
- K/F is a generalized Liouville extension iff the differential Galois group K over F has the solvable component of the identity
\(\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0\), \(a_i\in F\)
- E/F is generated by n linear independent solution to \(\mathcal{L}\), i.e. adjoining basis of \(V=\mathcal{L}^{-1}(0)\) to F
- \(C_E=C_F\), \(\partial_E\mid_F=\partial_F\)
- algebraic extension
- adjoining an integral
- adjoining the exponential of an integral
\(\operatorname{Gal}(E/F)=\{\sigma\in\operatorname{Aut}E|\partial(\sigma(n))=\sigma(\partial a), \sigma\mid _F=\operatorname{id} \}\)
- the action of an element of the Galois group is determined by its action on a basis of V
\(x(x-1)+px+q=0\)