Differential Galois theory - 편집 역사

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Pythagoras0: /* 메타데이터 */ 새 문단

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메타데이터: 새 문단

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imported>Pythagoras0: /* introduction */

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introduction

imported>Pythagoras0: /* books */

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books

imported>Pythagoras0: /* articles */

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articles

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articles

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 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q2383624 Q2383624]

← 이전 판		2020년 12월 28일 (월) 12:56 판
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	[[분류:math]]		[[분류:math]]
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q2383624 Q2383624]

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 * differential galois theory
 * Liouville
 * [[2008년 12월~09년 1월 한국 방문|2008년 12월]] 9일 MCF 'differential Galois theory'
-−
-−
 ==historical origin==
 * integration in finite terms
-* quadrature of second order differential equation (Fuchsian differential equation)
+* quadrature of second order differential equation (Fuchsian differential equation)
-−
-−
 ==solution by quadrature==
@@ 24번째 줄: / 24번째 줄: @@
 * note that the integral of an exponential naturally shows up in expression solutions
-−
-−
 ==differential field==
@@ 35번째 줄: / 35번째 줄: @@
 * <math>C_F=\ker \partial</math>
-−
-−
 ==solvable by quadratures==
 * basic functions : basic elementary functions
-* allowed operatrions : compositions, arithmetic operations, differentiation, integration
+* allowed operatrions : compositions, arithmetic operations, differentiation, integration
 *  examples
 ** an elliptic integral is representable by quadrature
-−
-−
 ==elementary extension==
@@ 58번째 줄: / 58번째 줄: @@
 ** 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
+* 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
@@ 73번째 줄: / 73번째 줄: @@
 **** exponential
 **** logarithm
-*  For<math>K_{i}=K_{i-1}(e_i)</math> , one of the following condition holds
+*  For<math>K_{i}=K_{i-1}(e_i)</math> , one of the following condition holds
 ** <math>e_i'\in K_{i-1}</math>, i.e. <math>e_i=\int e_i'\in K_i</math>
-** <math>e_{i}'/e_{i}\in K_{i-1}</math> i.e. <math>(\log e_i)' \in K_{i-1}</math>
+** <math>e_{i}'/e_{i}\in K_{i-1}</math> i.e. <math>(\log e_i)' \in K_{i-1}</math>
-** <math>e_{i}</math> is algebraic over <math>K_{i-1}</math>
+** <math>e_{i}</math> is algebraic over <math>K_{i-1}</math>
 *  remark on exponentiation
 **  Let <math>a,a'\in F</math>. Is <math>b=e^a\in K</math> where K is a Liouville extension?
@@ 86번째 줄: / 86번째 줄: @@
 *  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
+**  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)
+* 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<math>\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0</math>, <math>a_i\in F</math>
 * <math>(E,\partial_E)\supseteq (F,\partial_F)</math> is a Picard-Vessiot extension for <math>\mathcal{L}</math> if
@@ 112번째 줄: / 112번째 줄: @@
 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
+* differential equation with regular singularities
 *  indicial equation<math>x(x-1)+px+q=0</math>
 theorem
-A Fuchsian linear differential equation is solvable by quadratures if and only if the monodromy group of this equation is solvable.
+A Fuchsian linear differential equation is solvable by quadratures if and only if the monodromy group of this equation is solvable.
-−
-−
-−
 ==solution by quadrature==
@@ 139번째 줄: / 139번째 줄: @@
 * [http://www.math.purdue.edu/%7Eagabriel/topological_galois.pdf http://www.math.purdue.edu/~agabriel/topological_galois.pdf]
-−
-−
 ==related items==
@@ 149번째 줄: / 149번째 줄: @@
 * {{수학노트|url=푸크스_미분방정식(Fuchsian_differential_equation)}}
-−
-−
-−
 ==encyclopedia==
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 * http://en.wikipedia.org/wiki/Field_extension
-−
-−
 ==expositions==

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 * http://gigapedia.info/1/differntial+algebra
 [[분류:개인노트]]
 [[분류:math and physics]]
 [[분류:math]]
 [[분류:migrate]]

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 ==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]<math>\frac{dy}{dx}+a(x)y=b(x)</math><math>y(x)e^{\int a(x)\,dx}=\int b(x)e^{\int a(x)\,dx} \,dx+C</math>
-* <math>y''-2xy'=0</math><br><math>y=\int e^{x^2}\, dx</math><br>
+* <math>y''-2xy'=0</math><math>y=\int e^{x^2}\, dx</math>
 * note that the integral of an exponential naturally shows up in expression solutions
@@ 30번째 줄: / 30번째 줄: @@
 ==differential field==
-*  a pair <math>(F,\partial)</math> such that<br>
+*  a pair <math>(F,\partial)</math> such that
 ** <math>\partial(a+b)=\partial a+\partial b</math>
 ** <math>\partial(ab)=(\partial a)b+a(\partial b)</math>
@@ 43번째 줄: / 43번째 줄: @@
 * basic functions : basic elementary functions
 * allowed operatrions : compositions, arithmetic operations, differentiation, integration
-*  examples<br>
+*  examples
 ** an elliptic integral is representable by quadrature
@@ 54번째 줄: / 54번째 줄: @@
 * it is allowed to take exponentials and logarithms to make a field extension
 * elementary element
-*  difference between Liouville extension<br>
+*  difference between Liouville extension
 ** exponential+ integral <=> differentiation + exponential of integral
 ** in elementary extension, we are not allowed to get an integrated element
@@ 66번째 줄: / 66번째 줄: @@
 * 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<br>
+*  to get a Liouville extension, we can adjoin
 ** integrals
 ** exponentials of integrals
-**  algebraic extension (generalized Liouville extension)<br>
+**  algebraic extension (generalized Liouville extension)
-***  from these we can include the following operations<br>
+***  from these we can include the following operations
 **** exponential
 **** logarithm
-*  For<math>K_{i}=K_{i-1}(e_i)</math> , one of the following condition holds<br>
+*  For<math>K_{i}=K_{i-1}(e_i)</math> , one of the following condition holds
-** <math>e_i'\in K_{i-1}</math>, i.e. <math>e_i=\int e_i'\in K_i</math><br>
+** <math>e_i'\in K_{i-1}</math>, i.e. <math>e_i=\int e_i'\in K_i</math>
-** <math>e_{i}'/e_{i}\in K_{i-1}</math> i.e. <math>(\log e_i)' \in K_{i-1}</math><br>
+** <math>e_{i}'/e_{i}\in K_{i-1}</math> i.e. <math>(\log e_i)' \in K_{i-1}</math>
-** <math>e_{i}</math> is algebraic over <math>K_{i-1}</math><br>
+** <math>e_{i}</math> is algebraic over <math>K_{i-1}</math>
-*  remark on exponentiation<br>
+*  remark on exponentiation
-**  Let <math>a,a'\in F</math>. Is <math>b=e^a\in K</math> where K is a Liouville extension?<br>
+**  Let <math>a,a'\in F</math>. Is <math>b=e^a\in K</math> where K is a Liouville extension?
-** <math>b'=a' e^a=a'b</math> implies <math>a'=\frac{b'}{b}\in F</math>.<br>
+** <math>b'=a' e^a=a'b</math> implies <math>a'=\frac{b'}{b}\in F</math>.
-**  the exponential of the integral of a' i.e. <math>e^{\int a'}=e^a+c</math> must be in the Liouville extension. So <math>b=e^a\in K</math>.<br>
+**  the exponential of the integral of a' i.e. <math>e^{\int a'}=e^a+c</math> must be in the Liouville extension. So <math>b=e^a\in K</math>.
-*  remark on logarithm<br>
+*  remark on logarithm
-** <math>b=\log a</math> is the integral of <math>a'/a\in F</math>. So <math>b\in K</math><br>
+** <math>b=\log a</math> is the integral of <math>a'/a\in F</math>. So <math>b\in K</math>
-*  a few result<br>
+*  a few result
-**  K/F is a Liouville extension iff the differential Galois group K over F is solvable.<br>
+**  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<br>
+**  K/F is a generalized Liouville extension iff the differential Galois group K over F has the solvable component of the identity
@@ 96번째 줄: / 96번째 줄: @@
 * 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<br><math>\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0</math>, <math>a_i\in F</math><br>
+*  consider monic differential equations over a differential field F<math>\mathcal{L}[Y]=Y^{(n)}+a_{n-1}Y^{(n-1)}+\cdots+a_{1}Y^{(1)}+a_0</math>, <math>a_i\in F</math>
-* <math>(E,\partial_E)\supseteq (F,\partial_F)</math> is a Picard-Vessiot extension for <math>\mathcal{L}</math> if<br>
+* <math>(E,\partial_E)\supseteq (F,\partial_F)</math> is a Picard-Vessiot extension for <math>\mathcal{L}</math> if
 ** E/F is generated by n linear independent solution to <math>\mathcal{L}</math>, i.e. adjoining basis of <math>V=\mathcal{L}^{-1}(0)</math> to F
 ** <math>C_E=C_F</math>, <math>\partial_E\mid_F=\partial_F</math>
 * this corresponds to the concept of the splitting fields(or Galois extensions)
-*  examples<br>
+*  examples
 ** algebraic extension
 ** adjoining an integral
 ** adjoining the exponential of an integral
-*  we can define a Galois group for a linear differential equation<br><math>\operatorname{Gal}(E/F)=\{\sigma\in\operatorname{Aut}E|\partial(\sigma(n))=\sigma(\partial a), \sigma\mid _F=\operatorname{id} \}</math><br>
+*  we can define a Galois group for a linear differential equation<math>\operatorname{Gal}(E/F)=\{\sigma\in\operatorname{Aut}E|\partial(\sigma(n))=\sigma(\partial a), \sigma\mid _F=\operatorname{id} \}</math>
 ** the action of an element of the Galois group is determined by its action on a basis of V
@@ 119번째 줄: / 119번째 줄: @@
 * differential equation with regular singularities
-*  indicial equation<br><math>x(x-1)+px+q=0</math><br>
+*  indicial equation<math>x(x-1)+px+q=0</math>
 theorem
@@ 145번째 줄: / 145번째 줄: @@
 ==related items==
-* [[Class Field Theory]]<br>
+* [[Class Field Theory]]
-* [[number fields and threefolds]]<br>
+* [[number fields and threefolds]]
 * {{수학노트|url=푸크스_미분방정식(Fuchsian_differential_equation)}}
@@ 183번째 줄: / 183번째 줄: @@
 ==books==
-*  Group Theory and Differential Equations<br>
+*  Group Theory and Differential Equations
 ** Lawrence Markus, 1960
-*  An introduction to differential algebra<br>
+*  An introduction to differential algebra
-**  Irving Kaplansky<br>
+**  Irving Kaplansky
-*  algebraic theory of differential equations<br>
+*  algebraic theory of differential equations
 * http://gigapedia.info/1/galois_theory
 * http://gigapedia.info/1/differential+galois+theory

← 이전 판		2020년 11월 13일 (금) 06:40 판
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	* differential galois theory		* differential galois theory
	* Liouville		* Liouville
−	* [[2008년 12월~09년 1월 한국 방문\|2008년 12월]] 9일 MCF 'differential Galois theory'

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 ==articles==
 * Maurischat, Andreas. “A Categorical Approach to Picard-Vessiot Theory.” arXiv:1507.04166 [math], July 15, 2015. http://arxiv.org/abs/1507.04166.
 * Hardouin, Charlotte, and Alexey Ovchinnikov. ‘Calculating Galois Groups of Differential Equations with Parameters’. arXiv:1505.07068 [math], 26 May 2015. http://arxiv.org/abs/1505.07068.