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Difference between revisions of "User:Ulf Rehmann/Table of automatically generated TeX code"

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This page gives an analysis of [[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|the code here]], [[User:Maximilian Janisch/latexlist|generated automatically from some png files underlying our old wiki pages]].
 +
As this page does contain a lot of $\TeX$ code, it loads slowly.
 +
 +
Under the name of some of our EoM-pages the table below lists some png files, displaying their image and their $TeX$ rendering (automatically retrieved and corrected by hand).
 +
The first column gives the running number in this table, followed (in parentheses) by the number used [[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E| here]].
 +
The last column gives the confidence and the name of the png file, followed (in parentheses) by the number it has in the sequence of all png files called by its calling EoM-page.
 +
 +
Here is a short survey of the more systematic errors which seem to occur:
 +
 +
; 1. Trailing punctuation is dismissed.
 +
:[concerns almost all images] ; technically: pixels in sparse last pixel columns of bit images are suppressed/ignored?
 +
 +
; 2. "Displayed" images are not recognized as such.
 +
:[concerns almost all images]
 +
:Therefore these are displayed too small, and like "inline" $\TeX$ format.
 +
:
 +
:Remark: This cannot be discovered from the png file, it has to be retrieved from the html markup in the calling file: Displayed images are embedded in some html <table> markup.
 +
:
 +
;3. Sparse initial column pixels of the bit image are dismissed
 +
:(in parts this affects essential symbols), [see nr. 15,16,36,43,58,59,60,61,62,63,97,109]
 +
 +
;4. Some fonts are not recognized:
 +
:\cal:    [7.12.25.26,30,31,32,33,95,111] \mathbf: [30,83,111,127]  \bf:[ 133,148,149]
 +
:
 +
;5.  Semi-colon is interpreted as double pipe = "||" :[33,49,86,101]
 +
:
 +
;6.  Some code is not displayed at all.
 +
:    (This seems to be  a bug of our MathJax TeX interpreter.) [67,74,78,81,83,94,101,106]
 +
:    This seems to happen when a string "\text {" is involved, can apparently be fixed by using "\text{", but still unclear.
 +
:
 +
;7.  Questions:
 +
:    The different interpretation of the matrix delimiters in [56-63] is a bit surprising. Should be checked!
 +
:    Also, the vanishing of some '-' signs in the first column of some matrices, maybe that is related to 3.?
 +
 
==[[Algebraic curve]]==
 
==[[Algebraic curve]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 1.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|23.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011450/a01145065.png || $g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$ || $$ g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n, } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n, } \end{array} \right.$$ || conf 0.698
+
| 1.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|23.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011450/a01145065.png  
 +
|| $g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$  
 +
||$$g\leq \left\{
 +
\begin {array}{ll}
 +
{\frac {(n-2)^2}4} &{\text{ for even }n,}\\
 +
{\frac {(n-1)(n-3)}4} &{\text{ for odd }n,}
 +
\end {array}
 +
\right.$$
 +
|| conf 0.698
 
   
 
   
 
a01145065.png (65)
 
a01145065.png (65)
Line 17: Line 60:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 2.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|116.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150014.png || $\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || $$\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }, $$ || conf 0.997
+
| 2.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|116.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150014.png  
 +
|| $\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$  
 +
||$$\theta =\int\limits _ 0^{\lambda }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$
 +
|| conf 0.997
 
   
 
   
 
a01150014.png (14)
 
a01150014.png (14)
 
|-
 
|-
| 3.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|133.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150021.png || $\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || $$\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }, $$ || conf 0.973
+
| 3.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|133.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150021.png  
 +
|| $\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$  
 +
||$$\omega =2\int\limits _ 0^{1/c}\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$
 +
|| conf 0.973
 
   
 
   
 
a01150021.png (21)
 
a01150021.png (21)
 
|-
 
|-
| 4.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|67.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150022.png || $\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || $$\widetilde{ w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } },$$ || conf 0.107  
+
| 4.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|67.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150022.png  
 +
|| $\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$  
 +
||$$\widetilde w=2\int\limits _ 0^{1/\varepsilon }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$
 +
|| conf 0.107  
 
   
 
   
 
a01150022.png (22)
 
a01150022.png (22)
 
|-
 
|-
| 5.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|105.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150044.png || $\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ || $$\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v ), $$ || conf 0.775
+
| 5.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|105.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150044.png  
 +
|| $\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$  
 +
||$$\theta (v+\pi i r )=\theta (r),\quad \theta (v+\alpha _ j)=e^{L_j(v)}\theta (v),$$
 +
|| conf 0.775
 
   
 
   
 
a01150044.png (44)
 
a01150044.png (44)
 
|-
 
|-
| 6.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|17.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150078.png || $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ || $$\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 ). $$ || conf 0.440
+
| 6.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|17.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150078.png  
 +
|| $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$  
 +
||$$\left(
 +
\begin {array}{ll}
 +
{\alpha } &b\\
 +
c &d
 +
\end {array}
 +
\right)\equiv \left(
 +
\begin {array}{ll}
 +
1&0\\
 +
0&1
 +
\end {array}
 +
\right)(\operatorname {mod}7).$$
 +
|| conf 0.440
 
   
 
   
 
a01150078.png (78)
 
a01150078.png (78)
Line 48: Line 121:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 7.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|144.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640132.png || $0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$ || $$0 \rightarrow {\cal O} _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$$   || conf 0.981
+
| 7.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|144.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640132.png  
 +
|| $0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$  
 +
||$$0\rightarrow {\cal O}_V\rightarrow E _ {\alpha }\rightarrow T _ V\rightarrow 0$$
 +
|| conf 0.981
 
   
 
   
 
a011640132.png (132)
 
a011640132.png (132)
 
|-
 
|-
| 8.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|73.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640137.png  || $M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$ || $$ M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) ). $$|| conf 0.997
+
| 8.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|73.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640137.png   
 +
|| $M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$  
 +
||$$M=\operatorname {dim}\operatorname {Im}(H^1(V,E_{\alpha })\rightarrow H ^1(V,T_V)).$$
 +
|| conf 0.997
 
   
 
   
 
a011640137.png (137)
 
a011640137.png (137)
 
|-
 
|-
| 9.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|88.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640139.png  || $\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$ || $$ \operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } ). $$|| conf 0.996
+
| 9.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|88.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640139.png   
 +
|| $\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$  
 +
||$$\operatorname {dim}_kH^2(V,E_{\alpha })+\operatorname {dim}_kH^2(V,T_V).$$
 +
|| conf 0.996
 
   
 
   
 
a011640139.png (139)
 
a011640139.png (139)
 
|-
 
|-
| 10.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|117.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164027.png  || $N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$ || $$ N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1. $$|| conf 0.369
+
| 10.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|117.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164027.png   
 +
|| $N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$  
 +
||$$N_m=\left(\begin {array}c{m+3}\\
 +
3
 +
\end {array}
 +
\right)-dm+2t+\tau +p-1.$$
 +
|| conf 0.369
 
   
 
   
 
a01164027.png (27)
 
a01164027.png (27)
 
|-
 
|-
| 11.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|72.]]) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164029.png  || $p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$ || $$ p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1 $$|| conf 0.396
+
| 11.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|72.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164029.png   
 +
|| $p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$  
 +
||$$p_{\alpha }(V)=\left(\begin {array}c{n-1}\\
 +
3
 +
\end {array}
 +
\right)-d(n-1)+2t+\tau +p-1$$
 +
|| conf 0.396
 
   
 
   
 
a01164029.png (29)
 
a01164029.png (29)
 
|-
 
|-
| 12.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|68.]])*||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164047.png || $p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$ || $$p _ { \alpha } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , {\cal O} _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , {\cal O} _ { V } ) =$$ || conf 0.756  F  
+
| 12.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|68.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164047.png  
 +
|| $p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$  
 +
||$$p_{\alpha }(V)=-\operatorname {dim}_kH_1(V,{\cal O}_V)+\operatorname {dim}_kH^2(V,{\cal O}_V)=$$
 +
|| conf 0.756  F  
 
   
 
   
 
a01164047.png (47)
 
a01164047.png (47)
 
|-
 
|-
| 13.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|93.]])*||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164053.png || $1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$ || $$ 1 + p _ { \alpha } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 },$$|| conf 0.752  F  
+
| 13.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|93.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164053.png  
 +
|| $1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$  
 +
||$$1+p_{\alpha }(V)=\frac {\operatorname {deg}(c_1^2)+\operatorname {deg}(c_2)}{12},$$
 +
|| conf 0.752  F  
 
   
 
   
 
a01164053.png (53)
 
a01164053.png (53)
Line 87: Line 194:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 14.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|33.]])*||  https://www.encyclopediaofmath.org/legacyimages/c/c020/c020550/c0205509.png || $\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$ || $$\mathfrak { g }_0 = \big\{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists { n }_{X,H} \in {\mathbb Z} ( ( \text { ad } H ) ^ { n_{X , H} } ( X ) = 0 ) \big\}$$ || conf 0.110  F  
+
| 14.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|33.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/c/c020/c020550/c0205509.png  
 +
|| $\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$  
 +
||$$\mathfrak g_0=\big\{X\in \mathfrak g:\forall H \in \mathfrak t\exists n_{X,H}\in {\mathbb Z}((\text{ ad }H)^{n_{X,H}}(X)=0)\big\},$$
 +
|| conf 0.110  F  
  
 
c0205509.png (9)
 
c0205509.png (9)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Cartan theorem]]==
 
==[[Cartan theorem]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 15.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|49.]])*||  https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205704.png || $f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ || $$\empty$$ || conf 0.149  F  
+
| 15.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|49.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205704.png
 +
|| $f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$  
 +
||$$[e_i,f_j]=\delta _ {ij}h_i,\quad [h_i,e_j]=\alpha _ {ij}e_j,\quad [h_i,f_j]=-\alpha _ {ij}f_j,$$
 +
|| conf 0.149  F  
 
   
 
   
 
c0205704.png (4)
 
c0205704.png (4)
 
|-
 
|-
| 16.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|55.]])*||  https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057064.png || $\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$ || $$\empty$$ || conf 0.853  F  
+
| 16.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|55.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057064.png
 +
|| $\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$  
 +
||$$\dots \rightarrow H ^p(X,S)\rightarrow H ^p(X,F)\stackrel {\phi_p }{\rightarrow }H^p(X,G)\rightarrow $$
 +
|| conf 0.853  F  
 
   
 
   
 
c02057064.png (64)
 
c02057064.png (64)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Comitant]]==
 
==[[Comitant]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 17.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|7.]]) ||  https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333033.png || $H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ || $$\empty$$ || conf 0.956
+
| 17.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|7.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333033.png  
 +
|| $H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$  
 +
||$$H=\frac 1{36}\left|
 +
\begin {array}{cc}
 +
{\frac {\partial ^2f}{\partial x ^2}} &{\frac {\partial ^2f}{\partial x \partial y }}\\
 +
{\frac {\partial ^2f}{\partial x \partial y }} &{\frac {\partial ^2f}{\partial y ^2}}
 +
\end {array}
 +
\right|=$$
 +
|| conf 0.956
 
   
 
   
 
c02333033.png (33)
 
c02333033.png (33)
 
|-
 
|-
| 18.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|76.]]) ||  https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333034.png || $= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$ || $$\empty$$ || conf 0.549
+
| 18.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|76.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333034.png
 +
|| $= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$  
 +
||$$=(a_0a_2-a_1^2)x^2+(a_0a_3-a_1a_2)xy+(a_1a_3-a_2^2)y^2$$
 +
|| conf 0.549
 
   
 
   
 
c02333034.png (34)
 
c02333034.png (34)
 
|-
 
|-
| 19.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|11.]])*||  https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333035.png || $( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ || $$\empty$$ || conf 0.521  F  
+
| 19.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|11.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333035.png
 +
|| $( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$  
 +
||$$(\alpha _ 0,\alpha _ 1,\alpha _ 2,\alpha _ 3)\mapsto (\alpha _ 0\alpha _ 2-\alpha _ 1^2,\frac 12(\alpha _ 0\alpha _ 3-\alpha _ 1\alpha _ 2),\alpha _ 1\alpha _ 3-\alpha _ 2^2)$$
 +
|| conf 0.521  F  
 
   
 
   
 
c02333035.png (35)
 
c02333035.png (35)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Deformation]]==
 
==[[Deformation]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 20.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|26.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700175.png || $\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ || $$\empty$$ || conf 0.683
+
| 20.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|26.]])  
   
+
||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700175.png  
 +
|| $\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$  
 +
||$$\operatorname {Aut}_{R^{\prime }}(X^{\prime }|X_0)\rightarrow \operatorname {Aut}_R(X_{R^{\prime }}^{\prime }\otimes R |X_0)$$
 +
|| conf 0.683
 +
  \
 
d030700175.png (175)
 
d030700175.png (175)
 
|-
 
|-
| 21.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|27.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700190.png || $\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$ || $$\empty$$ || conf 0.944
+
| 21.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|27.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700190.png
 +
|| $\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$  
 +
||$$\operatorname {dim}_kH^1(X_0,T_{X_0})-\operatorname {dim}M_{X_0}\leq \operatorname {dim}_kH^2(X_0,T_{X_0}).$$
 +
|| conf 0.944
 
   
 
   
 
d030700190.png (190)
 
d030700190.png (190)
 
|-
 
|-
| 22.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|78.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700263.png || $\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$ || $$\empty$$ || conf 0.097  F  
+
| 22.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|78.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700263.png
 +
|| $\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$  
 +
||$$\alpha \circ b =\alpha b +\sum _ {i=1}^{\infty }\phi _ i(\alpha ,b)t^i,\quad \alpha ,b\in V,$$
 +
|| conf 0.097  F  
 
   
 
   
 
d030700263.png (263)
 
d030700263.png (263)
 
|-
 
|-
| 23.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|96.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700270.png || $\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$ || $$\empty$$ || conf 0.873  F  
+
| 23.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|96.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700270.png
 +
|| $\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$  
 +
||$$\Phi (\alpha )=\alpha +\sum _ {i=1}^{\infty }t^i\phi _ i(\alpha ),\quad \alpha \in V,$$
 +
|| conf 0.873  F  
 
   
 
   
 
d030700270.png (270)
 
d030700270.png (270)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Differential algebra]]==
 
==[[Differential algebra]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 24.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|106.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830107.png || $S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ || $$\empty$$ || conf 0.149
+
| 24.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|106.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830107.png
 +
|| $S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$  
 +
||$$S^tF=\sum _ {j=1}^rc_jA^{p_j}A_1^{i_{1j}}\dots A _ {m-l}^{i_{{m-l},j}},$$
 +
|| conf 0.149
 
   
 
   
 
d031830107.png (107)
 
d031830107.png (107)
 
|-
 
|-
| 25.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|146.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830141.png || $( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$ || $$\empty$$ || conf 0.562  F  
+
| 25.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|146.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830141.png
 +
|| $( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$  
 +
||$(\eta _ 1,\ldots ,\eta _ k)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ k)$
 +
|| conf 0.562  F  
 
   
 
   
 
d031830141.png (141)
 
d031830141.png (141)
 
|-
 
|-
| 26.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|145.]])$^F$*||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830150.png || $( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$ || $$\empty$$ || conf 0.376  F  
+
| 26.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|145.]])$^F$*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830150.png
 +
|| $( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$  
 +
||$(\eta _ 1,\ldots ,\eta _ n)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ n)$
 +
|| conf 0.376  F  
 
   
 
   
 
d031830150.png (150)
 
d031830150.png (150)
 
|-
 
|-
| 27.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|57.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d03183016.png || $\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ || $$\empty$$ || conf 0.780
+
| 27.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|57.]])  
 +
 
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d03183016.png
 +
 
 +
|| $\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$  
 +
 
 +
||$$\omega _ V=\sum _ {0\leq i \leq m }\alpha _ i\left(
 +
\begin {array}c{x+i}\\
 +
i
 +
\end {array}
 +
\right),$$
 +
|| conf 0.780
 
   
 
   
 
d03183016.png (16)
 
d03183016.png (16)
 
|-
 
|-
| 28.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|111.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d03183043.png || $e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$ || $$ e_ { i j } = \operatorname { ord } _ { { Y } _ { j } } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$$ || conf 0.187  
+
| 28.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|111.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d03183043.png  
 +
|| $e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$  
 +
||$$e_{ij}=\operatorname {ord}_{Y_j}F_i,\quad 1 \leq i \leq n ,\quad i \leq j \leq n,$$
 +
|| conf 0.187  
 
   
 
   
 
d03183043.png (43)
 
d03183043.png (43)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Dimension polynomial]]==
 
==[[Dimension polynomial]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 29.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|48.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249029.png || $\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ || $$\empty$$ || conf 0.968
+
| 29.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|48.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249029.png  
 +
|| $\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$  
 +
||$$\omega _ {\eta /F}(x)=\sum _ {0\leq i \leq m }\alpha _ i\left(\begin {array}c{x+i}\\
 +
i
 +
\end {array}
 +
\right),$$
 +
|| conf 0.968
 
   
 
   
 
d03249029.png (29)
 
d03249029.png (29)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Duality]]==
 
==[[Duality]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 30.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|118.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120173.png || $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ || $$\empty$$ || conf 0.824  F  
+
| 30.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|118.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120173.png
 +
|| $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$  
 +
||$$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow {\mathbf C},$$
 +
|| conf 0.824  F  
 
   
 
   
 
d034120173.png (173)
 
d034120173.png (173)
 
|-
 
|-
| 31.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|59.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120175.png || $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$ || $$\empty$$ || conf 0.921  F  
+
| 31.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|59.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120175.png
 +
|| $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$  
 +
||$$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow H _ c^n(X,\Omega )$$
 +
|| conf 0.921  F  
 
   
 
   
 
d034120175.png (175)
 
d034120175.png (175)
 
|-
 
|-
| 32.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|124.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120184.png || $( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$ || $$\empty$$ || conf 0.829  F  
+
| 32.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|124.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120184.png
 +
|| $( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$  
 +
||$$(H^p(X,{\cal F}))^{\prime }\cong H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega )).$$
 +
|| conf 0.829  F  
 
   
 
   
 
d034120184.png (184)
 
d034120184.png (184)
 
|-
 
|-
| 33.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|29.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120236.png || $\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X  F  , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y || F , \Omega )$ || $$\empty$$ || conf 0.634 || F
+
| 33.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|29.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120236.png
 +
|| $\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X  F  , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y  
 +
|| F , \Omega )$  
 +
||$$\beta :\operatorname {Ext}_c^{n-p-1}(X;{\cal F},\Omega )\rightarrow \operatorname {Ext}_c^{n-p-1}(X\backslash Y ;{\cal F},\Omega ).$$
 +
|| conf 0.634  
 +
|| F
 
   
 
   
 
d034120236.png (236)
 
d034120236.png (236)
 
|-
 
|-
| 34.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|77.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120247.png || $\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$ || $$\empty$$ || conf 0.521  F  
+
| 34.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|77.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120247.png
 +
|| $\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$  
 +
||$$\underset {n\rightarrow \infty }{\overline {\lim }}|\alpha _ n|^{1/n}=\sigma <+\infty.$$
 +
|| conf 0.521  F  
 
   
 
   
 
d034120247.png (247)
 
d034120247.png (247)
 
|-
 
|-
| 35.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|58.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120253.png || $h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$ || $$\empty$$ || conf 0.861  F  
+
| 35.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|58.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120253.png
 +
|| $h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$  
 +
||$$h(\phi )=\underset {n\rightarrow \infty }{\overline {\lim }}\frac {\operatorname {ln}|A(re^{i\phi })|}r$$
 +
|| conf 0.861  F  
 
   
 
   
 
d034120253.png (253)
 
d034120253.png (253)
 
|-
 
|-
| 36.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|69.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120360.png || $\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$ || $$\empty$$ || conf 0.293  F  
+
| 36.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|69.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120360.png
 +
|| $\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$  
 +
||$$\operatorname*{sup}_{l\in E^\perp \atop \|l\|\le 1 }|l(\omega )|=\operatorname*{inf}_{x\in E }\|\omega -x\|,$$
 +
|| conf 0.293  F  
 
   
 
   
 
d034120360.png (360)
 
d034120360.png (360)
 
|-
 
|-
| 37.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|15.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120376.png || $\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$ || $$\empty$$ || conf 0.508
+
| 37.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|15.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120376.png
 +
|| $\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$  
 +
||$$\operatorname*{sup}_{f\in B ^1}\big|\int\limits _ {\partial G }f(\zeta )\omega (\zeta )d\zeta \big|=\operatorname*{inf}_{\phi \in E ^1}\int\limits _ {\partial G }|\omega (\zeta )-\phi (\zeta )
 +
||d\zeta |.$$
 +
|| conf 0.508
 
   
 
   
 
d034120376.png (376)
 
d034120376.png (376)
 
|-
 
|-
| 38.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|52.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120509.png || $f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ || $$\empty$$ || conf 0.491
+
| 38.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|52.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120509.png
 +
|| $f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$  
 +
||$$f=\{f_{\alpha }\}\in \prod _ {\alpha }F_{\alpha },\quad g =\{g_{\alpha }\}\in \operatorname*\oplus _ {\alpha }G_{\alpha }.$$
 +
|| conf 0.491
 
   
 
   
 
d034120509.png (509)
 
d034120509.png (509)
 
|-
 
|-
| 39.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|140.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120535.png || $f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$ || $$\empty$$ || conf 0.900
+
| 39.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|140.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120535.png
 +
|| $f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$  
 +
||$$f^{*}(x^{*})=\operatorname*{sup}_{x\in X }(\langle x ^{*},x\rangle -f(x))$$
 +
|| conf 0.900
 
   
 
   
 
d034120535.png (535)
 
d034120535.png (535)
 
|-
 
|-
| 40.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|94.]]) ||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120555.png || $f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$ || $$\empty$$ || conf 0.810
+
| 40.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|94.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120555.png
 +
|| $f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$  
 +
||$$f_0(x)\rightarrow \text{ inf, }\quad f _ i(x)\leq 0 ,\quad i =1,\ldots ,m,\quad x \in B,$$
 +
|| conf 0.810
 
   
 
   
 
d034120555.png (555)
 
d034120555.png (555)
 
|-
 
|-
| 41.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|74.]])*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d03412079.png || $( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ || $$\empty$$ || conf 0.117  F  
+
| 41.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|74.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d03412079.png
 +
|| $( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$  
 +
||$$(c_{\gamma },c^r)=\sum _ {t^r\in K }c_r(t^{\prime })c^r(t^r)\operatorname {mod}1$$
 +
|| conf 0.117  F  
 
   
 
   
 
d03412079.png (79)
 
d03412079.png (79)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Extension of a differential field]]==
 
==[[Extension of a differential field]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
!style=width: 7%| Confidence, F?
+
!style=width: 7%| Confidence, F?  
 
png file  
 
png file  
 
|-
 
|-
| 42.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|63.]]) ||  https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696024.png || $F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ || $$\empty$$ || conf 0.628
+
| 42.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|63.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696024.png  
 +
|| $F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$  
 +
||$$F_1F_2=F_1\langle F _ 2\rangle =F_1(F_2)=F_2(F_1)=F_2\langle F _ 1\rangle,$$
 +
|| conf 0.628
 
   
 
   
 
e03696024.png (24)
 
e03696024.png (24)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Formal group]]==
 
==[[Formal group]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 43.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|120.]])*||  https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f040820118.png || $\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ || $$\empty$$ || conf 0.098  F  
+
| 43.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|120.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f040820118.png  
 +
|| $\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$  
 +
||$$\operatorname {log}F_{\rm MU }(X)=\sum _ {i=1}^{\infty }i^{-1}[{\rm CP}^{i-1}]X^i,$$
 +
|| conf 0.098  F  
 
   
 
   
 
f040820118.png (118)
 
f040820118.png (118)
 
|-
 
|-
| 44.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|147.]])*||  https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f04082059.png || $( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ || $$\empty$$ || conf 0.553  F  
+
| 44.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|147.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f04082059.png  
 +
|| $( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$  
 +
||$$(x_1,\ldots ,x_n)\circ (y_1,\ldots ,y_n)=(z_1,\ldots ,z_n),$$
 +
|| conf 0.553  F  
 
   
 
   
 
f04082059.png (59)
 
f04082059.png (59)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Gel'fond-Schneider method]]==
 
==[[Gel'fond-Schneider method]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 45.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|148.]]) ||  https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300205.png || $\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$ || $$\empty$$ || conf 0.979
+
| 45.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|148.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300205.png  
 +
|| $\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$  
 +
||$\alpha ^{\beta }=\operatorname {exp}\{\beta \operatorname {log}\alpha \}$
 +
|| conf 0.979
 
   
 
   
 
g1300205.png (5)
 
g1300205.png (5)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Group]]==
 
==[[Group]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 46.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|22.]])*||  https://www.encyclopediaofmath.org/legacyimages/g/g045/g045210/g04521075.png || $\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$ || $$\empty$$ || conf 0.226  F  
+
| 46.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|22.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/g/g045/g045210/g04521075.png  
 +
|| $\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$  
 +
| style="text-align:center;"| source incomplete
 +
|| conf 0.226  F  
 
   
 
   
 
g04521075.png (75)
 
g04521075.png (75)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Homogeneous space]]==
 
==[[Homogeneous space]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 47.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|89.]]) ||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047690/h04769069.png || $\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$ || $$\empty$$ || conf 0.793
+
| 47.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|89.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047690/h04769069.png  
 +
|| $\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$  
 +
||$$\mathfrak g=\mathfrak f+\mathfrak m,\quad \mathfrak f\cap \mathfrak m=\{0\},$$
 +
|| conf 0.793
 
   
 
   
 
h04769069.png (69)
 
h04769069.png (69)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Hopf algebra]]==
 
==[[Hopf algebra]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 48.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|103.]]) ||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970129.png || $m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$ || $$\empty$$ || conf 0.618
+
| 48.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|103.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970129.png  
 +
|| $m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$  
 +
||$m\circ (\iota \otimes 1 )\circ \mu =m\circ (1\otimes \iota )\circ \mu =e\circ \epsilon$
 +
|| conf 0.618
 
   
 
   
 
h047970129.png (129)
 
h047970129.png (129)
 
|-
 
|-
| 49.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|107.]])*||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970139.png || $F _ { 1 } ( X || Y ) , \ldots , F _ { n } ( X || Y ) \in K [ X _ { 1 } , \ldots , X _ { n } || Y _ { 1 } , \ldots , Y _ { n } ] \}$ || $$\empty$$ || conf 0.353  F  
+
| 49.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|107.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970139.png  
 +
|| $F _ { 1 } ( X  
 +
|| Y ) , \ldots , F _ { n } ( X  
 +
|| Y ) \in K [ X _ { 1 } , \ldots , X _ { n }  
 +
|| Y _ { 1 } , \ldots , Y _ { n } ] \}$  
 +
||$F_1(X;Y),\ldots ,F_n(X;Y)\in K [X_1,\ldots ,X_n;Y_1,\ldots ,Y_n]\}$
 +
|| conf 0.353  F  
 
   
 
   
 
h047970139.png (139)
 
h047970139.png (139)
 
|-
 
|-
| 50.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|97.]]) ||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797042.png || $\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$ || $$\empty$$ || conf 0.213
+
| 50.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|97.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797042.png  
 +
|| $\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$  
 +
||$$\epsilon (x)=0,\quad \delta (x)=x\otimes 1 +1\otimes x ,\quad x \in \mathfrak g.$$
 +
|| conf 0.213
 
   
 
   
 
h04797042.png (42)
 
h04797042.png (42)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Invariants, theory of]]==
 
==[[Invariants, theory of]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 51.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|149.]])*||  https://www.encyclopediaofmath.org/legacyimages/i/i052/i052350/i05235015.png || $\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ || $$\empty$$ || conf 0.142  F  
+
| 51.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|149.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/i/i052/i052350/i05235015.png  
 +
|| $\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$  
 +
||$$\alpha _ {i_1,\dots,i_n}\rightarrow \alpha _ {i_1,\dots,i_n}^{\prime }.$$
 +
|| conf 0.142  F  
 
   
 
   
 
i05235015.png (15)
 
i05235015.png (15)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Jordan algebra]]==
 
==[[Jordan algebra]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 52.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|150.]]) ||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427030.png || $H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ || $$\empty$$ || conf 0.651
+
| 52.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|150.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427030.png  
 +
|| $H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$  
 +
||$$(C_3,\Gamma )=\big\{X\in C _ 3:X=\Gamma ^{-1}X\square ^{\prime }\Gamma \big\},$$
 +
|| conf 0.651
 
   
 
   
 
j05427030.png (30)
 
j05427030.png (30)
 
|-
 
|-
| 53.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|42.]]) ||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427031.png || $\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$ || $$\empty$$ || conf 0.987  
+
| 53.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|42.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427031.png  
 +
|| $\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$  
 +
||$$\Gamma =\operatorname {diag}\{\gamma _ 1,\gamma _ 2,\gamma _ 3\},\quad \gamma _ i\neq 0 ,\quad \gamma _ i\in F,$$
 +
|| conf 0.987  
 
   
 
   
 
j05427031.png (31)
 
j05427031.png (31)
 
|-
 
|-
| 54.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|125.]])*||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427077.png || $\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$ || $$\empty$$ || conf 0.598  F  
+
| 54.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|125.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427077.png  
 +
|| $\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$  
 +
||$\mathfrak g=\mathfrak g_{-1}+\mathfrak g_0+\mathfrak g_1$
 +
|| conf 0.598  F  
 
   
 
   
 
j05427077.png (77)
 
j05427077.png (77)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Jordan matrix]]==
 
==[[Jordan matrix]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 55.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|6.]])*||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543403.png || $J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$ || $$J = \left\| \begin{array} { c c c c }  
+
| 55.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|6.]])*
  J_{n_1}(\lambda_1) &   0     &   0   & 0 \\  
+
||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543403.png  
      0             & \ddots & \ddots & 0 \\
+
|| $J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$  
      0             & \ddots & \ddots & 0 \\
+
||$$J=\left\|
      0             &   0     &   0   & J_{n_s}(\lambda_s)
+
\begin {array}{cccc}
\end{array} \right\|,$$ || conf 0.072  F  
+
 +
  J_{n_1}(\lambda_1) &0 &0 &0\\
 +
 +
0 &\ddots &\ddots &0\\
 +
 +
0 &\ddots &\ddots &0\\
 +
 +
0 &0 &0 &J_{n_s}(\lambda_s)
 +
 +
\end {array}
 +
\right\|,$$
 +
|| conf 0.072  F  
 
   
 
   
 
j0543403.png (3)
 
j0543403.png (3)
 
|-
 
|-
| 56.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|64.]]) ||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434030.png || $C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ || $$\empty$$ || conf 0.955
+
| 56.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|64.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434030.png
 +
|| $C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$  
 +
||$$C_m(\lambda )=\operatorname {rk}(A-\lambda E )^{m-1}-2\operatorname {rk}(A-\lambda E )^m+$$
 +
|| conf 0.955
 
   
 
   
 
j05434030.png (30)
 
j05434030.png (30)
 
|-
 
|-
| 57.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|1.]])*||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543406.png || $J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$ || $$J_m(\lambda) = \left\| \begin{array} { c c c c c c }
+
| 57.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|1.]])*
\lambda &   1   & \square & \square & \square & \square \\
+
||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543406.png  
\square & \lambda &   1   & \square &   0     & \square \\
+
|| $J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$  
\square & \square & \ddots & \ddots & \square & \square\\
+
||$$J_m(\lambda)=\left\|
\square & \square & \square & \ddots & \ddots & \square \\
+
\begin {array}{cccccc}
\square &   0     & \square & \square & \lambda & 1 \\
+
\square & \square & \square & \square & \square & \lambda
+
\lambda &1 &\square &\square &\square &\square \\
\end{array} \right\|,$$ || conf 0.098  F  
+
 +
\square &\lambda &1 &\square &0 &\square \\
 +
 +
\square &\square &\ddots &\ddots &\square &\square\\
 +
 +
\square &\square &\square &\ddots &\ddots &\square \\
 +
 +
\square &0 &\square &\square &\lambda &1\\
 +
 +
\square &\square &\square &\square &\square &\lambda  
 +
\end {array}
 +
\right\|,$$
 +
|| conf 0.098  F  
 
   
 
   
 
j0543406.png (6)
 
j0543406.png (6)
Line 435: Line 769:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 58.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|5.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510127.png || $\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ || $$B_n: \quad \left\| \begin{array} { r r r r r r }
+
| 58.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|5.]])  
{ 2 & { - 1 } & { 0 & { \dots } & { 0 } & { 0 \\  
+
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510127.png  
{ - 1 } & { 2 & { - 1 } & { \dots } & { 0 } & { 0 \\  
+
|| $\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$  
{ 0 & { - 1 } & { 2 & { \dots } & { 0 } & { 0 \\  
+
||$$B_n:\quad \left\|
  \cdot & \cdot   & \cdot   &   \dots   & \cdot & \cdot \\
+
\begin {array}{rrrrrr}
{ 0 & { 0 & { 0 & { \dots } & { - 1 } & { 0 }\\  
+
{ 0 & { 0 & { 0 & { \dots } & { 2 } & { - 2 }\\  
+
2 &{-1} &0 &{\dots } &0 &0\\
{ 0 & { 0 & { 0 & { \dots } & { - 1 } & { 2 }
+
\end{array} \right\|,$$|| conf 0.232
+
{-1} &2 &{-1} &{\dots } &0 &0\\
 +
 +
0 &{-1} &2 &{\dots } &0 &0\\
 +
 +
  \cdot &\cdot &\cdot &\dots &\cdot &\cdot \\
 +
 +
0 &0 &0 &{\dots } &{-1} &0\\
 +
 +
0 &0 &0 &{\dots } &2 &{-2}\\
 +
 +
0 &0 &0 &{\dots } &{-1} &2
 +
 +
\end {array}
 +
\right\|,$$
 +
|| conf 0.232
 
   
 
   
 
l058510127.png (127)
 
l058510127.png (127)
 
|-
 
|-
| 59.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|3.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510129.png || $\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ || $$D_n: \quad  
+
| 59.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|3.]])*
\left\| \begin{array} { r r r r r r r }  
+
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510129.png  
{ 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\  
+
|| $\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$  
{ - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 }\\  
+
||$$D_n:\quad \left\|
{ 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 \\  
+
\begin {array}{rrrrrrr}
\cdot & \cdot   & \cdot &   \dots   & \cdot & \cdot &\cdot & \cdot \\
+
{ 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 \\  
+
2 &{-1} &0 &{\dots } &0 &0 &0 &0\\
{ 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 }\\  
+
{ 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\  
+
{-1} &2 &{-1} &{\dots } &0 &0 &0 &0\\
{ 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 }
+
\end{array} \right\|,$$ || conf 0.055  F  
+
0 &{-1} &2 &{\dots } &0 &0 &0 &0\\
 +
 +
\cdot &\cdot &\cdot &\dots &\cdot &\cdot &\cdot &\cdot \\
 +
 +
0 &0 &0 &{\dots } &2 &{-1} &0 &0\\
 +
 +
0 &0 &0 &{\dots } &{-1} &2 &{-1} &{-1}\\
 +
 +
0 &0 &0 &{\dots } &0 &{-1} &2 &0\\
 +
 +
0 &0 &0 &{\dots } &0 &{-1} &0 &2
 +
 +
\end {array}
 +
\right\|,$$
 +
|| conf 0.055  F  
  
 
l058510129.png (129)
 
l058510129.png (129)
 
|-
 
|-
| 60.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|8.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510130.png || $\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ || $$E_6:  
+
| 60.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|8.]])*
\quad \left\| \begin{array} { r r r r r r }  
+
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510130.png  
{ 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\  
+
|| $\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$  
{ 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\  
+
||$$E_6:
{ - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\  
+
\quad \left\|
{ 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\  
+
\begin {array}{rrrrrr}
{ 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\  
+
{ 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 }
+
2 &0 &{-1} &0 &0 &0\\
\end{array} \right\|,$$ || conf 0.628  F
+
 +
0 &2 &0 &{-1} &0 &0\\
 +
 +
{-1} &0 &2 &{-1} &0 &0\\
 +
 +
0 &{-1} &{-1} &2 &{-1} &0\\
 +
 +
0 &0 &0 &{-1} &2 &{-1}\\
 +
 +
0 &0 &0 &0 &{-1} &2
 +
 +
\end {array}
 +
\right\|,$$
 +
|| conf 0.628  F
  
 
l058510130.png (130)
 
l058510130.png (130)
 
|-
 
|-
| 61.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|4.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510131.png || $\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ || $$E_7: \quad  
+
| 61.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|4.]])  
\left\| \begin{array} { r r r r r r r }  
+
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510131.png  
{ 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\  
+
|| $\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$  
{ 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\  
+
||$$E_7:\quad \left\|
{-1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\  
+
\begin {array}{rrrrrrr}
{ 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\  
+
{ 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\  
+
2 &0 &{-1} &0 &0 &0 &0\\
{ 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\  
+
{ 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 }
+
0 &2 &0 &{-1} &0 &0 &0\\
\end{array} \right\|,$$ || conf 0.278
+
 +
{-1} &0 &2 &{-1} &0 &0 &0\\
 +
 +
0 &{-1} &{-1} &2 &{-1} &0 &0\\
 +
 +
0 &0 &0 &{-1} &2 &{-1} &0\\
 +
 +
0 &0 &0 &0 &{-1} &2 &{-1}\\
 +
 +
0 &0 &0 &0 &0 &{-1} &2
 +
 +
\end {array}
 +
\right\|,$$
 +
|| conf 0.278
 
   
 
   
 
l058510131.png (131)
 
l058510131.png (131)
 
|-
 
|-
| 62.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|2.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510132.png || $\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ || $$E_8: \quad  
+
| 62.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|2.]])*
\left\| \begin{array} { r r r r r r r r }  
+
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510132.png  
{ 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } &  
+
|| $\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$  
{ 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\  
+
||$$E_8:\quad \left\|
{-1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\  
+
\begin {array}{rrrrrrrr}
{ 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\  
+
{ 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\  
+
2 &0 &{-1} &0 &0 &0 &0 &
{ 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\  
+
0\\
{ 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\  
+
0 &2 &0 &{-1} &0 &0 &0 &0\\
{ 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 }
+
\end{array} \right\|,$$ || conf 0.354  F  
+
{-1} &0 &2 &{-1} &0 &0 &0 &0\\
 +
 +
0 &{-1} &{-1} &2 &{-1} &0 &0 &0\\
 +
 +
0 &0 &0 &{-1} &2 &{-1} &0 &0\\
 +
 +
0 &0 &0 &0 &{-1} &2 &{-1} &0\\
 +
 +
0 &0 &0 &0 &0 &{-1} &2 &{-1}\\
 +
 +
0 &0 &0 &0 &0 &0 &{-1} &2
 +
 +
\end {array}
 +
\right\|,$$
 +
|| conf 0.354  F  
 
   
 
   
 
l058510132.png (132)
 
l058510132.png (132)
 
|-
 
|-
| 63.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|10.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510133.png || $\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$  
+
| 63.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|10.]])*
|| $$F_4: \quad \left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|.$$ || conf 0.374  F  
+
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510133.png  
 +
|| $\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$  
 +
 
 +
||$$F_4:\quad \left\|
 +
\begin {array}{rrrr}
 +
2 &{-1} &0 &0\\
 +
{-1} &2 &{-2} &0\\
 +
0 &{-1} &2 &{-1}\\
 +
0 &0 &{-1} &2
 +
\end {array}
 +
\right\|,\quad G _ 2:\quad \left\|
 +
\begin {array}{rr}
 +
2&{-1}\\
 +
{-3}&2
 +
\end {array}
 +
\right\|.$$
 +
|| conf 0.374  F  
 
   
 
   
 
l058510133.png (133)
 
l058510133.png (133)
 
|-
 
|-
| 64.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|98.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851030.png || $\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$ || $$\empty$$ || conf 0.976
+
| 64.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|98.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851030.png
 +
|| $\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$  
 +
||$$\mathfrak g_{\alpha }=\{X\in \mathfrak g:[H,X]=\alpha (H)X,H\in \mathfrak h\}.$$
 +
|| conf 0.976
 
   
 
   
 
l05851030.png (30)
 
l05851030.png (30)
 
|-
 
|-
| 65.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|126.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851037.png || $\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ || $$\empty$$ || conf 0.945
+
| 65.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|126.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851037.png
 +
|| $\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$  
 +
||$$\mathfrak g=\mathfrak h+\sum _ {\alpha \in \Sigma }\mathfrak g_{\alpha }.$$
 +
|| conf 0.945
 
   
 
   
 
l05851037.png (37)
 
l05851037.png (37)
 
|-
 
|-
| 66.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|61.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851044.png || $\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$ || $$\empty$$ || conf 0.520  F  
+
| 66.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|61.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851044.png
 +
|| $\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$  
 +
||$$\mathfrak g_{\alpha }=\operatorname {dim}[\mathfrak g_{\alpha },\mathfrak g_{-\alpha }]=1.$$
 +
|| conf 0.520  F  
 
   
 
   
 
l05851044.png (44)
 
l05851044.png (44)
 
|-
 
|-
| 67.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|65.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851050.png || $[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ || $$\empty$$ || conf 0.539  F  
+
| 67.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|65.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851050.png
 +
|| $[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$  
 +
||$$[H_{\alpha },X_{\alpha }]=2X_{\alpha }\quad {\rm and }\quad [H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }.$$
 +
|| conf 0.539  F  
 
   
 
   
 
l05851050.png (50)
 
l05851050.png (50)
 
|-
 
|-
| 68.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|70.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851051.png || $\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$ || $$\empty$$ || conf 0.997
+
| 68.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|70.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851051.png
 +
|| $\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$  
 +
||$$\beta (H_{\alpha })=\frac {2(\alpha ,\beta )}{(\alpha ,\alpha )},\quad \alpha ,\beta \in \Sigma,$$
 +
|| conf 0.997
 
   
 
   
 
l05851051.png (51)
 
l05851051.png (51)
 
|-
 
|-
| 69.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|112.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851057.png || $[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ || $$\empty$$ || conf 0.917
+
| 69.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|112.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851057.png
 +
|| $[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$  
 +
||$$[\mathfrak g_{\alpha },\mathfrak g_{\beta }]=\mathfrak g_{\alpha +\beta }$$
 +
|| conf 0.917
 
   
 
   
 
l05851057.png (57)
 
l05851057.png (57)
 
|-
 
|-
| 70.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|127.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851064.png || $H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$ || $$\empty$$ || conf 0.432
+
| 70.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|127.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851064.png
 +
|| $H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$  
 +
||$$H_{\alpha _ 1},\ldots ,H_{\alpha _ k},X_{\alpha }\quad (\alpha \in \Sigma )$$
 +
|| conf 0.432
 
   
 
   
 
l05851064.png (64)
 
l05851064.png (64)
 
|-
 
|-
| 71.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|113.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851069.png || $[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ || $$\empty$$ || conf 0.628  F  
+
| 71.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|113.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851069.png
 +
|| $[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$  
 +
||$$[[X_{\alpha _ i},X_{-}\alpha _ i],X_{-\alpha _ j}]=-n(i,j)X_{\alpha _ j},$$
 +
|| conf 0.628  F  
 
   
 
   
 
l05851069.png (69)
 
l05851069.png (69)
 
|-
 
|-
| 72.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|79.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851073.png || $n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$ || $$\empty$$ || conf 0.992
+
| 72.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|79.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851073.png
 +
|| $n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$  
 +
||$$n(i,j)=\alpha _ j(H_i)=\frac {2(\alpha _ i,\alpha _ j)}{(\alpha _ j,\alpha _ j)}.$$
 +
|| conf 0.992
 
   
 
   
 
l05851073.png (73)
 
l05851073.png (73)
 
|-
 
|-
| 73.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|13.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851074.png || $[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ || $$\empty$$ || conf 0.988
+
| 73.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|13.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851074.png
 +
|| $[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$  
 +
||$$[X_{\alpha },X_{\beta }]=\left\{
 +
\begin {array}{ll}
 +
{N_{\alpha ,\beta }X_{\alpha +\beta }} &{\text{ if }\alpha +\beta \in \Sigma,}\\
 +
0 &{\text{ if }\alpha +\beta \notin \Sigma,}
 +
\end {array}
 +
\right.$$
 +
|| conf 0.988
 
   
 
   
 
l05851074.png (74)
 
l05851074.png (74)
 
|-
 
|-
| 74.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|80.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851078.png || $N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$ || $$\empty$$ || conf 0.961
+
| 74.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|80.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851078.png
 +
|| $N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$  
 +
||$$N_{\alpha ,\beta }=-N_{-\alpha ,-\beta }\quad {\rm and }\quad N _ {\alpha ,\beta }=\pm (p+1),$$
 +
|| conf 0.961
 
   
 
   
 
l05851078.png (78)
 
l05851078.png (78)
 
|-
 
|-
| 75.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|85.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851085.png || $X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ || $$\empty$$ || conf 0.691  F  
+
| 75.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|85.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851085.png
 +
|| $X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$  
 +
||$$iH_\alpha,X_{\alpha }-X_{-\alpha },\quad i (X_{\alpha }+X_{-\alpha })\quad (\alpha \in \Sigma _ {+})$$
 +
|| conf 0.691  F  
 
   
 
   
 
l05851085.png (85)
 
l05851085.png (85)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Lie algebra, solvable]]==
 
==[[Lie algebra, solvable]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 76.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|119.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852011.png || $[ \mathfrak { g } _ { i } , \mathfrak { g } _ { i } ] \subset \mathfrak { g } _ { \mathfrak { i } } + 1$ || $$\empty$$ || conf 0.276  F  
+
| 76.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|119.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852011.png  
 +
|| $[ \mathfrak { g } _ { i } , \mathfrak { g } _ { i } ] \subset \mathfrak { g } _ { \mathfrak { i } } + 1$  
 +
||$[\mathfrak g_i,\mathfrak g_i]\subset \mathfrak g_{i+1}$
 +
|| conf 0.276  F  
 
   
 
   
 
l05852011.png (11)
 
l05852011.png (11)
 
|-
 
|-
| 77.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|141.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852046.png || $\operatorname { dim } \mathfrak { g } _ { i } = \operatorname { dim } \mathfrak { g } - i$ || $$\empty$$ || conf 0.901
+
| 77.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|141.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852046.png  
 +
|| $\operatorname { dim } \mathfrak { g } _ { i } = \operatorname { dim } \mathfrak { g } - i$  
 +
||$\operatorname {dim}\mathfrak g_i=\operatorname {dim}\mathfrak g-i$
 +
|| conf 0.901
 
   
 
   
 
l05852046.png (46)
 
l05852046.png (46)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Lie group]]==
 
==[[Lie group]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 78.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|62.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l058590115.png || $( G ) \cong \operatorname { Aut } ( L ( G ) ) \quad \text { and } \quad L ( \operatorname { Aut } ( G ) ) \cong D ( L ( G ) )$ || $$\empty$$ || conf 0.693  F  
+
| 78.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|62.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l058590115.png  
 +
|| $( G ) \cong \operatorname { Aut } ( L ( G ) ) \quad \text { and } \quad L ( \operatorname { Aut } ( G ) ) \cong D ( L ( G ) )$  
 +
||$$\operatorname {Aut}(G)\cong \operatorname {Aut}(L(G))\quad {\rm and }\quad L (\operatorname {Aut}(G))\cong D (L(G)),$$
 +
|| conf 0.693  F  
 
   
 
   
 
l058590115.png (115)
 
l058590115.png (115)
 
|-
 
|-
| 79.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|50.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l05859086.png || $( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ || $$\empty$$ || conf 0.856
+
| 79.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|50.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l05859086.png  
 +
|| $( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$  
 +
||$$(X,Y)\rightarrow \operatorname {exp}^{-1}(\operatorname {exp}X\operatorname {exp}Y),\quad X ,Y\in L (G),$$
 +
|| conf 0.856
 
   
 
   
 
l05859086.png (86)
 
l05859086.png (86)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Lie group, compact]]==
 
==[[Lie group, compact]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 80.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|121.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058610/l05861012.png || $J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$ || $$\empty$$ || conf 0.364  F  
+
| 80.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|121.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058610/l05861012.png  
 +
|| $J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$  
 +
||$$J=\left\|
 +
\begin {array}{cc}
 +
0 &{E_x}\\
 +
{-E_x} &0
 +
\end {array}
 +
\right\|,$$
 +
|| conf 0.364  F  
 
   
 
   
 
l05861012.png (12)
 
l05861012.png (12)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Lie group, nilpotent]]==
 
==[[Lie group, nilpotent]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 81.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|83.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586604.png || $N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$ || $$\empty$$ || conf 0.466
+
| 81.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|83.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586604.png  
 +
|| $N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$  
 +
||$$N(F)=\{g\in GL (V):gv\equiv v \operatorname {mod}V_i\;\text{for all }v\in V _ i,\;i\geq 1 \}$$
 +
|| conf 0.466
 
   
 
   
 
l0586604.png (4)
 
l0586604.png (4)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Lie group, semi-simple]]==
 
==[[Lie group, semi-simple]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 82.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|35.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680102.png || $L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$ || $$\empty$$ || conf 0.659  F  
+
| 82.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|35.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680102.png
 +
|| $L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$  
 +
||$$L(\mathfrak g)\cong \Gamma _ 0(\mathfrak u)\cap \mathfrak h^{\prime }/\Gamma _ 0([\mathfrak k,\mathfrak k])$$
 +
|| conf 0.659  F  
 
   
 
   
 
l058680102.png (102)
 
l058680102.png (102)
 
|-
 
|-
| 83.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|81.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868032.png || $\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ || $$\empty$$ || conf 0.183  F  
+
| 83.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|81.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868032.png
 +
|| $\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$  
 +
||$$\Gamma _ 1=\Gamma _ 1(g)=\{X\in h :\alpha (X)\in 2 \pi i {\mathbf Z}\;\text{for all }\alpha \in \Sigma \}.$$
 +
|| conf 0.183  F  
 
   
 
   
 
l05868032.png (32)
 
l05868032.png (32)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Lie p-algebra]]==
 
==[[Lie p-algebra]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 84.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|36.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872026.png || $( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$ || $$\empty$$ || conf 0.356
+
| 84.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|36.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872026.png
 +
|| $( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$  
 +
||$$(\operatorname {ad}x)^ny=\sum _ {j=1}^n(-1)^j\left(\begin {array}cn\\
 +
j
 +
\end {array}
 +
\right)x^{n-j}yx^j$$
 +
|| conf 0.356
 
   
 
   
 
l05872026.png (26)
 
l05872026.png (26)
 
|-
 
|-
| 85.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|99.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872078.png || $\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ || $$\empty$$ || conf 0.964
+
| 85.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|99.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872078.png
 +
|| $\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$  
 +
||$$\pi (x+y)=\pi (x)+\pi (y),\quad \pi (\lambda x )=\lambda ^p\pi (x),\quad \lambda \in k .$$
 +
|| conf 0.964
 
   
 
   
 
l05872078.png (78)
 
l05872078.png (78)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Lie theorem]]==
 
==[[Lie theorem]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 86.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|134.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876010.png || $y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ || $$\empty$$ || conf 0.276
+
| 86.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|134.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876010.png
 +
|| $y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i }  
 +
|| x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$  
 +
||$$y_i=f_i(g_1,\ldots ,g_i;x_1,\ldots ,x_n),\quad i =1,\ldots ,n$$
 +
|| conf 0.276
 
   
 
   
 
l05876010.png (10)
 
l05876010.png (10)
 
|-
 
|-
| 87.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|86.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876016.png || $X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ || $$\empty$$ || conf 0.656
+
| 87.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|86.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876016.png
 +
|| $X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$  
 +
||$$X_i=\sum _ {j=1}^n\xi _ {ij}(x)\frac {\partial }{\partial x _ j},\quad i =1,\ldots ,r,$$
 +
|| conf 0.656
 
   
 
   
 
l05876016.png (16)
 
l05876016.png (16)
 
|-
 
|-
| 88.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|66.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876030.png || $\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$ || $$\empty$$ || conf 0.336  F  
+
| 88.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|66.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876030.png
 +
|| $\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$  
 +
||$$\frac {\partial f _ j}{\partial g _ i}(g,x)=\sum _ {k=1}^r\xi _ {kj}(f(g_sx))\psi _ {ki}(g),$$
 +
|| conf 0.336  F  
 
   
 
   
 
l05876030.png (30)
 
l05876030.png (30)
 
|-
 
|-
| 89.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|19.]])*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876037.png || $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ || $$\empty$$ || conf 0.157  F  
+
| 89.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|19.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876037.png
 +
|| $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$  
 +
||$$\sum _ {k=1}^N(\xi _ {ik}\frac {\partial \xi _ {jl}}{\partial x _ k}-\xi _ {jk}\frac {\partial \xi _ {il}}{\partial x _ k})=\sum _ {k=1}^rc_{ij}^k\xi _ {kl},$$
 +
|| conf 0.157  F  
 
   
 
   
 
l05876037.png (37)
 
l05876037.png (37)
 
|-
 
|-
| 90.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|14.]]) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876052.png || $\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ || $$\empty$$ || conf 0.085
+
| 90.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|14.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876052.png
 +
|| $\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$  
 +
||$$\left.\begin {array}c{c_{ij}^k=-c_{ji}^k},\\
 +
{\displaystyle\sum _ {l=1}^r(c_{il}^mc_{jk}^l+c_{kl}^mc_{ij}^l+c_{jl}^mc_{ki}^l)=0,\quad 1 \leq i ,j,k,l,m\leq r,}
 +
\end {array}
 +
\right\}$$
 +
|| conf 0.085
 
   
 
   
 
l05876052.png (52)
 
l05876052.png (52)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Maximal torus]]==
 
==[[Maximal torus]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 91.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|95.]]) ||  https://www.encyclopediaofmath.org/legacyimages/m/m063/m063010/m06301072.png || $F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$ || $$\empty$$ || conf 0.198
+
| 91.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|95.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/m/m063/m063010/m06301072.png  
 +
|| $F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$  
 +
||$$F(x_1f_1+\ldots +x_xf_n)=x_1x_n+x_2x_{n-1}+\ldots +x_px_{n-p+1},$$
 +
|| conf 0.198
 
   
 
   
 
m06301072.png (72)
 
m06301072.png (72)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Non-Abelian cohomology]]==
 
==[[Non-Abelian cohomology]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
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!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 92.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|114.]])*||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900110.png || $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ || $$\empty$$ || conf 0.443  F  
+
| 92.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|114.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900110.png
 +
|| $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$  
 +
||$$\phi (g_1)\phi (g_2)\phi (g_1g_2)^{-1}=\operatorname {Int}m(g_1,g_2),$$
 +
|| conf 0.443  F  
 
   
 
   
 
n066900110.png (110)
 
n066900110.png (110)
 
|-
 
|-
| 93.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|90.]])*||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900118.png || $( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$ || $$\empty$$ || conf 0.764  F  
+
| 93.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|90.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900118.png
 +
|| $( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$  
 +
||$$m'(g_1,g_2)=h(g_1)(\phi (g_1)(h(g_2)))m(g_1,g_2)h(g_1,g_2)^{-1}.$$
 +
|| conf 0.764  F  
 
   
 
   
 
n066900118.png (118)
 
n066900118.png (118)
 
|-
 
|-
| 94.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|44.]]) ||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690016.png || $\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ || $$\empty$$ || conf 0.400
+
| 94.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|44.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690016.png
 +
|| $\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$  
 +
||$$\delta (e)=e\quad \;\text{and }\quad \delta (\rho (a)b)=\sigma (a)\delta (b),\quad \alpha \in C ^0,\quad b \in C ^1,$$
 +
|| conf 0.400
 
   
 
   
 
n06690016.png (16)
 
n06690016.png (16)
 
|-
 
|-
| 95.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|60.]])*||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690028.png || $C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$ || $$\empty$$ || conf 0.205  F  
+
| 95.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|60.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690028.png
 +
|| $C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$  
 +
||$$C^{*}(\mathfrak U,{\cal F})=(C^0(\mathfrak U,{\cal F}),C^1(\mathfrak U,{\cal F}),C^2(\mathfrak U,{\cal F})),$$
 +
|| conf 0.205  F  
 
   
 
   
 
n06690028.png (28)
 
n06690028.png (28)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Picard scheme]]==
 
==[[Picard scheme]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
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!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 96.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|39.]])*||  https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267025.png || $\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$ || $$\empty$$ || conf 0.345  F +
+
| 96.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|39.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267025.png
 +
|| $\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$  
 +
||$$\operatorname {Pic}_{X/k}(S^{\prime })=\operatorname {Pic}(X\times_k S ^{\prime })/\operatorname {Pic}(S^{\prime })$$
 +
|| conf 0.345  F +
 
   
 
   
 
p07267025.png (25)
 
p07267025.png (25)
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!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 97.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|100.]])*||  https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464025.png || $g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$ || $$\empty$$ || conf 0.184  F  
+
| 97.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|100.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464025.png
 +
|| $g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$  
 +
||$$g_j:U_i\cap U _ j\rightarrow G ,\quad i ,j\in I ,\quad U _ i\cap U _ j\neq \emptyset,$$
 +
|| conf 0.184  F  
 
   
 
   
 
p07464025.png (25)
 
p07464025.png (25)
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!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 98.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|101.]]) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631062.png || $\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$ || $$\empty$$ || conf 0.837
+
| 98.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|101.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631062.png
 +
|| $\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$  
 +
||$$\phi ^{*}:\mathfrak g^{*}\otimes \mathfrak g^{*}\rightarrow \mathfrak g^{*}$$
 +
|| conf 0.837
 
   
 
   
 
q07631062.png (62)
 
q07631062.png (62)
 
|-
 
|-
| 99.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|108.]]) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631071.png || $\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ || $$\empty$$ || conf 0.648
+
| 99.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|108.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631071.png
 +
|| $\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$  
 +
||$$\delta :U_{\mathfrak g}\rightarrow U _ {\mathfrak g}\otimes U _ {\mathfrak g}$$
 +
|| conf 0.648
 
   
 
   
 
q07631071.png (71)
 
q07631071.png (71)
 
|-
 
|-
| 100.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|56.]])*||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631072.png || $\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ || $$\empty$$ || conf 0.304  F  
+
| 100.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|56.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631072.png
 +
|| $\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$  
 +
||$$\delta (\alpha )=\operatorname {lim}_{h\rightarrow 0 }h^{-1}(\Delta (a)-\Delta ^{\prime }(\alpha ))$$
 +
|| conf 0.304  F  
 
   
 
   
 
q07631072.png (72)
 
q07631072.png (72)
 
|-
 
|-
| 101.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|129.]])*||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631088.png || $[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$ || $$\empty$$ || conf 0.544  F  
+
| 101.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|129.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631088.png
 +
|| $[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$  
 +
||$$[\alpha ,X_i^{\pm }]=\pm \alpha _ i(\alpha )X_i^{\pm }\quad \text{for }a\in \mathfrak h;$$
 +
|| conf 0.544  F  
 
   
 
   
 
q07631088.png (88)
 
q07631088.png (88)
 
|-
 
|-
| 102.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|128.]]) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631089.png || $[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ || $$\empty$$ || conf 0.893
+
| 102.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|128.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631089.png
 +
|| $[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$  
 +
||$$[X_i^{+},X_j^{-}]=2\delta _ {ij}h^{-1}\operatorname {sinh}(hH_i/2).$$
 +
|| conf 0.893
 
   
 
   
 
q07631089.png (89)
 
q07631089.png (89)
 
|-
 
|-
| 103.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|20.]]) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631092.png || $\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$ || $$\empty$$ || conf 0.055
+
| 103.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|20.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631092.png
 +
|| $\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$  
 +
||$$\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\
 +
k
 +
\end {array}
 +
\right)q^{-k(n-k)/2}(X_i^{\pm })^kX_j^{\pm }\cdot (X_i^{\pm })^{n-k}=0.$$
 +
|| conf 0.055
 
   
 
   
 
q07631092.png (92)
 
q07631092.png (92)
 
|-
 
|-
| 104.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|30.]]) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631095.png || $\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ || $$\empty$$ || conf 0.443
+
| 104.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|30.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631095.png
 +
|| $\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$  
 +
||$$\left(
 +
\begin {array}ln\\
 +
k
 +
\end {array}
 +
\right)_q=\frac {(q^n-1)\ldots (q^{n-k+1}-1)}{(q^k-1)\ldots (q-1)}
 +
.$$
 +
|| conf 0.443
 
   
 
   
 
q07631095.png (95)
 
q07631095.png (95)
 
|-
 
|-
| 105.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|21.]])*||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631099.png || $\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$ || $$\empty$$ || conf 0.212  F  
+
| 105.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|21.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631099.png
 +
|| $\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$  
 +
||$$\Delta (X_i^{\pm })=X_i^{\pm }\otimes \operatorname {exp}(\frac {hH_i}4)+\operatorname {exp}(\frac {-hH_i}4)\otimes X _ i^{\pm }.$$
 +
|| conf 0.212  F  
 
   
 
   
 
q07631099.png (99)
 
q07631099.png (99)
Line 803: Line 1,424:
 
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+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 106.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|91.]]) ||  https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630100.png || $0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$ || $$\empty$$ || conf 0.879
+
| 106.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|91.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630100.png
 +
|| $0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$  
 +
||$$0\leq \frac {2(\chi ,\alpha )}{(\alpha ,\alpha )}<p\quad \text{for all }\alpha \in \Delta.$$
 +
|| conf 0.879
 
   
 
   
 
r077630100.png (100)
 
r077630100.png (100)
 
|-
 
|-
| 107.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|135.]]) ||  https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630104.png || $\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ || $$\empty$$ || conf 0.136
+
| 107.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|135.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630104.png
 +
|| $\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$  
 +
||$$\phi _ 0\otimes \phi _ 1^{Fr}\otimes \ldots \otimes \phi _ d^{{Fr}^d},$$
 +
|| conf 0.136
 
   
 
   
 
r077630104.png (104)
 
r077630104.png (104)
 
|-
 
|-
| 108.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|45.]])*||  https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763055.png || $\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$ || $$\empty$$ || conf 0.862  F  
+
| 108.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|45.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763055.png
 +
|| $\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$  
 +
||$$\chi =\delta _ {\phi }-\sum _ {\alpha \in \Delta }m_{\alpha }\alpha ,\quad m _ {\alpha }\in Z ,\quad m _ {\alpha }\geq 0.$$
 +
|| conf 0.862  F  
 
   
 
   
 
r07763055.png (55)
 
r07763055.png (55)
Line 825: Line 1,458:
 
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+
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+
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!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
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|-
 
|-
| 109.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|31.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590225.png || $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ || $$\empty$$ || conf 0.324
+
| 109.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|31.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590225.png
 +
|| $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$  
 +
||$$\sum _ {k_1,\ldots ,k_n=0}^{\infty }c_{k_1\cdots k _ n}(z_1-\zeta _ 1)^{k_1}\ldots (z_n-\zeta _ n)^{k_n}$$
 +
|| conf 0.324
 
   
 
   
 
s085590225.png (225)
 
s085590225.png (225)
 
|-
 
|-
| 110.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|46.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590404.png || $\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ || $$\empty$$ || conf 0.459
+
| 110.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|46.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590404.png
 +
|| $\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$  
 +
||$$\frac {m_1}{n_1}<\frac {m_2}{n_1n_2}<\ldots <\frac {m_g}{n_1\ldots n _ g}=\frac {m_g}n$$
 +
|| conf 0.459
 
   
 
   
 
s085590404.png (404)
 
s085590404.png (404)
 
|-
 
|-
| 111.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|115.]])*||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590429.png || $p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ || $$\empty$$ || conf 0.997  F  
+
| 111.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|115.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590429.png
 +
|| $p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$  
 +
||$$p(Z)=1-\operatorname {dim}H^0({\mathbf Z},{\cal O}_{\mathbf Z })+\operatorname {dim}H^1({\mathbf Z},{\cal O}_{\mathbf Z })$$
 +
|| conf 0.997  F  
 
   
 
   
 
s085590429.png (429)
 
s085590429.png (429)
 
|-
 
|-
| 112.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|136.]])*||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590440.png || $X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$ || $$\empty$$ || conf 0.433  F  
+
| 112.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|136.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590440.png
 +
|| $X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$  
 +
||$$X_{\epsilon }=\{(x_0,\ldots ,x_x):f(x_0,\ldots ,x_x)=\epsilon \}$$
 +
|| conf 0.433  F  
 
   
 
   
 
s085590440.png (440)
 
s085590440.png (440)
 
|-
 
|-
| 113.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|12.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590458.png || $= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ || $$\empty$$ || conf 0.870
+
| 113.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|12.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590458.png
 +
|| $= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$  
 +
||$$=\left\{
 +
\begin {array}{ll}
 +
{(x+\lambda )^2\ldots (x+k\lambda )^2} &{\text{ if }\mu =2k,}\\
 +
{(x+\lambda )^2\ldots (x+k\lambda )^2(x+(k+1)\lambda )} &{\text{ if }\mu =2k+1,}
 +
\end {array}
 +
\right.$$
 +
|| conf 0.870
 
   
 
   
 
s085590458.png (458)
 
s085590458.png (458)
 
|-
 
|-
| 114.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|75.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590482.png || $( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ || $$\empty$$ || conf 0.986
+
| 114.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|75.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590482.png
 +
|| $( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$  
 +
||$$\big(\frac {\partial F (x,y,\lambda )}{\partial x },\frac {\partial F (x,y,\lambda )}{\partial y }\big)$$
 +
|| conf 0.986
 
   
 
   
 
s085590482.png (482)
 
s085590482.png (482)
 
|-
 
|-
| 115.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|137.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590515.png || $\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$ || $$\empty$$ || conf 0.594
+
| 115.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|137.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590515.png
 +
|| $\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$  
 +
||$$\frac {dx_i}{dx_{i_0}}=f_i(x),\quad f _ i\in C (U),\quad i \neq i _ 0.$$
 +
|| conf 0.594
 
   
 
   
 
s085590515.png (515)
 
s085590515.png (515)
 
|-
 
|-
| 116.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|142.]])*||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590527.png || $A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ || $$\empty$$ || conf 0.506  F  
+
| 116.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|142.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590527.png
 +
|| $A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$  
 +
||$$A=\left\|
 +
\begin {array}{ll}
 +
{\alpha } &b\\
 +
c &e
 +
\end {array}
 +
\right\|$$
 +
|| conf 0.506  F  
 
   
 
   
 
s085590527.png (527)
 
s085590527.png (527)
 
|-
 
|-
| 117.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|53.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590634.png || $\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$ || $$\empty$$ || conf 0.920
+
| 117.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|53.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590634.png
 +
|| $\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$  
 +
||$$\Delta =(F_{xx}^{\prime \prime })_0(F_{yy}^{\prime \prime })_0-(F_{xy}^{\prime \prime })_0^2$$
 +
|| conf 0.920
 
   
 
   
 
s085590634.png (634)
 
s085590634.png (634)
 
|-
 
|-
| 118.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|16.]])*||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590645.png || $\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ || $$\empty$$ || conf 0.230  F  
+
| 118.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|16.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590645.png
 +
|| $\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$  
 +
||$$\left\|
 +
\begin {array}{lll}
 +
{F_x^{\prime }} &{F_y^{\prime }} &{F_z^{\prime }}\\
 +
{G_x^{\prime }} &{G_y^{\prime }} &{G_Z^{\prime }}
 +
\end {array}
 +
\right\|$$
 +
|| conf 0.230  F  
 
   
 
   
 
s085590645.png (645)
 
s085590645.png (645)
 
|-
 
|-
| 119.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|92.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590653.png || $( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$ || $$\empty$$ || conf 0.300
+
| 119.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|92.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590653.png
 +
|| $( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$  
 +
||$$(F_x^{\prime })_0=0,\quad (F_y^{\prime })_0=0,\quad (F_z^{\prime })_0=0.$$
 +
|| conf 0.300
 
   
 
   
 
s085590653.png (653)
 
s085590653.png (653)
Line 879: Line 1,571:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 120.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|138.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610054.png || $\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$ || $$\empty$$ || conf 0.972
+
| 120.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|138.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610054.png
 +
|| $\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$  
 +
||$$\{e\}\rightarrow \Delta \rightarrow \pi \rightarrow {\mathbf Z}^s\rightarrow \{e\}$$
 +
|| conf 0.972
 
   
 
   
 
s08610054.png (54)
 
s08610054.png (54)
Line 893: Line 1,589:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 121.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|71.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706033.png || $\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$ || $$\empty$$ || conf 0.379
+
| 121.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|71.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706033.png
 +
|| $\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$  
 +
||$$\psi _ {t_1,\ldots ,t_n}^{\prime }:SK_1(R)\rightarrow S K _ 1(R(t_1,\ldots ,t_n)).$$
 +
|| conf 0.379
 
   
 
   
 
s08706033.png (33)
 
s08706033.png (33)
Line 907: Line 1,607:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 122.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|130.]]) ||  https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053016.png || $e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$ || $$\empty$$ || conf 0.138
+
| 122.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|130.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053016.png
 +
|| $e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$  
 +
||$$e=\frac {|U|}{|G|}\big(\sum _ {b\in B }b\big)\big(\sum _ {w\in W }\operatorname {sign}(w)w\big)$$
 +
|| conf 0.138
 
   
 
   
 
s13053016.png (16)
 
s13053016.png (16)
Line 921: Line 1,625:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 123.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|24.]])*||  https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054017.png || $( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ || $$\empty$$ || conf 0.381  F  
+
| 123.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|24.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054017.png
 +
|| $( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$  
 +
||$$(x_{ij}(a),x_{kl}(b))=\left\{
 +
\begin {array}{ll}
 +
1 &{\text{ if }i\neq l ,j\neq k },\\
 +
{x_{il}(ab)} &{\text{ if }i\neq l ,j=k}.
 +
\end {array}
 +
\right.$$
 +
|| conf 0.381  F  
 
   
 
   
 
s13054017.png (17)
 
s13054017.png (17)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Tilting theory]]==
 
==[[Tilting theory]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 124.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|84.]]) ||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130105.png || $0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$ || $$\empty$$ || conf 0.946
+
| 124.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|84.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130105.png
 +
|| $0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$  
 +
||$$0\rightarrow \Lambda \rightarrow T _ 1\rightarrow \ldots \rightarrow T _ n\rightarrow 0 $$
 +
|| conf 0.946
 
   
 
   
 
t130130105.png (105)
 
t130130105.png (105)
Line 949: Line 1,667:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 125.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|18.]]) ||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140104.png || $q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ || $$\empty$$ || conf 0.112
+
| 125.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|18.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140104.png
 +
|| $q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$  
 +
||$$q_R(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}x_ix_j+\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}r_{i,j}x_ix_j,$$
 +
|| conf 0.112
 
   
 
   
 
t130140104.png (104)
 
t130140104.png (104)
 
|-
 
|-
| 126.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|40.]]) ||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140118.png || $[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$ || $$\empty$$ || conf 0.116  
+
| 126.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|40.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140118.png
 +
|| $[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$  
 +
||$$[X]\mapsto \chi _ R([X])=\sum _ {m=0}^{\infty }(-1)^m\operatorname {dim}_K\operatorname {Ext}_R^m(X,X)$$
 +
|| conf 0.116  
 
   
 
   
 
t130140118.png (118)
 
t130140118.png (118)
 
|-
 
|-
| 127.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|132.]])*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140119.png || $\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ || $$\empty$$ || conf 0.287 F  
+
| 127.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|132.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140119.png
 +
|| $\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$  
 +
||$$\underline {\dim }:K_0(\operatorname {mod}R)\rightarrow {\mathbf Z}^{Q_0}$$
 +
|| conf 0.287 F  
 
   
 
   
 
t130140119.png (119)
 
t130140119.png (119)
 
|-
 
|-
| 128.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|37.]])*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140140.png || $q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ || $$\empty$$ || conf 0.197  F  
+
| 128.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|37.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140140.png
 +
|| $q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$  
 +
||$$q_I(x)=\sum _ {i\in I }x_i^2+\sum _ {i\prec j \atop j\in I\setminus {\rm max}I}x_ix_j-\sum _ {p\in \operatorname {max}I}\big(\sum _ {i\prec p }x_i\big)x_p$$
 +
|| conf 0.197  F  
 
   
 
   
 
t130140140.png (140)
 
t130140140.png (140)
 
|-
 
|-
| 129.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|131.]])*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014044.png || $X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ || $$\empty$$ || conf 0.819  F  
+
| 129.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|131.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014044.png
 +
|| $X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$  
 +
||$$X\mapsto \underline {\dim }X=(\operatorname {dim}_KX_j)_{j\in Q _ 0}$$
 +
|| conf 0.819  F  
 
   
 
   
 
t13014044.png (44)
 
t13014044.png (44)
 
|-
 
|-
| 130.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|25.]]) ||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014048.png || $[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$ || $$\empty$$ || conf 0.661
+
| 130.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|25.]]  
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014048.png
 +
|| $[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$  
 +
||$$[X]\mapsto \chi _ Q([X])=\operatorname {dim}_K\operatorname {End}_Q(X)-\operatorname {dim}_K\operatorname {Ext}_Q^1(X,X)$$
 +
|| conf 0.661
 
   
 
   
 
t13014048.png (48)
 
t13014048.png (48)
 
|-
 
|-
| 131.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|38.]])*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014056.png || $A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ || $$\empty$$ || conf 0.481  F  
+
| 131.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|38.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014056.png
 +
|| $A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$  
 +
||$$A_Q(v)=\prod _ {i,j\in Q _ 0}\prod _ {\langle \beta :j\rightarrow i \rangle \in Q _ 1}M_{v_i\times v _ j}(K)_{\beta }$$
 +
|| conf 0.481  F  
 
   
 
   
 
t13014056.png (56)
 
t13014056.png (56)
 
|-
 
|-
| 132.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|139.]])*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301406.png || $\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ || $$\empty$$ || conf 0.648  F  
+
| 132.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|139.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301406.png
 +
|| $\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$  
 +
||$$q_Q(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {i,j\in Q _ 0}d_{ij}x_ix_j,$$
 +
|| conf 0.648  F  
 
   
 
   
 
t1301406.png (6)
 
t1301406.png (6)
 
|-
 
|-
 
|}
 
|}
 +
 
==[[Torus]]==
 
==[[Torus]]==
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
{| class="wikitable" style="text-align: left; width: 1740px;"
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 133.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|41.]])*||  https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933502.png || $r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ || $$\empty$$ || conf 0.585  F  
+
| 133.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|41.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933502.png
 +
|| $r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$  
 +
||$$r=\alpha \operatorname {sin}u{\bf k}+l(1+\epsilon \operatorname {cos}u)({\bf i}\operatorname {cos}v+{\bf j}\operatorname {sin}v)$$
 +
|| conf 0.585  F  
 
   
 
   
 
t0933502.png (2)
 
t0933502.png (2)
 
|-
 
|-
| 134.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|122.]])*||  https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933507.png || $d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$ || $$\empty$$ || conf 0.696  F  
+
| 134.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|122.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933507.png
 +
|| $d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$  
 +
||$$ds^2=\alpha ^2du^2+l^2(1+\epsilon \operatorname {cos}u)^2dv^2,$$
 +
|| conf 0.696  F  
 
   
 
   
 
t0933507.png (7)
 
t0933507.png (7)
Line 1,009: Line 1,768:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 135.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|9.]]) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524027.png || $u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ || $$\empty$$ || conf 0.733
+
| 135.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|9.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524027.png
 +
|| $u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$  
 +
||$$u_3(x)=\left\{
 +
\begin {array}{ll}
 +
{\frac {x^2}2,} &{0\leq x <1,}\\
 +
{\frac {[x^2-3(x-1)^2]}2,} &{1\leq x <2,}\\
 +
{\frac {[x^2-3(x-1)^2+3(x-2)^2]}2,} &{2\leq x <3,}\\
 +
{0,} &{x\notin [0,3].}
 +
\end {array}
 +
\right.$$
 +
|| conf 0.733
 
   
 
   
 
u09524027.png (27)
 
u09524027.png (27)
 
|-
 
|-
| 136.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|32.]])*||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952403.png || $p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$ || $$\empty$$ || conf 0.681  F  
+
| 136.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|32.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952403.png
 +
|| $p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$  
 +
||$$p(x)=\left\{
 +
\begin {array}{ll}
 +
{\frac 1{b-\alpha },} &{x\in [\alpha ,b],}\\
 +
{0,} &{x\notin [\alpha ,b].}
 +
\end {array}
 +
\right.$$
 +
|| conf 0.681  F  
 
   
 
   
 
u0952403.png (3)
 
u0952403.png (3)
 
|-
 
|-
| 137.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|34.]]) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524030.png || $u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ || $$\empty$$ || conf 0.569
+
| 137.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|34.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524030.png
 +
|| $u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$  
 +
||$$u_n(x)=\frac 1{(n-1)!}\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\
 +
k
 +
\end {array}
 +
\right)(x-k)_{+}^{n-1}$$
 +
|| conf 0.569
 
   
 
   
 
u09524030.png (30)
 
u09524030.png (30)
 
|-
 
|-
| 138.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|109.]]) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524034.png || $z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$ || $$\empty$$ || conf 0.676
+
| 138.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|109.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524034.png
 +
|| $z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$  
 +
||$$z_{+}=\left\{
 +
\begin {array}{ll}
 +
{z,} &{z>0}.\\
 +
{0,} &{z\leq 0 }.
 +
\end {array}
 +
\right.$$
 +
|| conf 0.676
 
   
 
   
 
u09524034.png (34)
 
u09524034.png (34)
 
|-
 
|-
| 139.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|43.]]) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952407.png || $F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ || $$\empty$$ || conf 0.468
+
| 139.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|43.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952407.png
 +
|| $F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$  
 +
||$$F(x)=\left\{
 +
\begin {array}{ll}
 +
{0,} &{x\leq a },\\
 +
{\frac {x-a}{b-a},} &{a<x\leq b },\\
 +
{1,} &{x>b},
 +
\end {array}
 +
\right.$$
 +
|| conf 0.468
 
   
 
   
 
u0952407.png (7)
 
u0952407.png (7)
 
|-
 
|-
| 140.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|47.]]) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524072.png || $p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$ || $$\empty$$ || conf 0.705
+
| 140.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|47.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524072.png
 +
|| $p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$  
 +
||$$p(x_1,\ldots ,x_n)=\left\{
 +
\begin {array}{ll}
 +
{C\neq 0 ,} &{x\in D },\\
 +
{0,} &{x\notin D },
 +
\end {array}
 +
\right.$$
 +
|| conf 0.705
 
   
 
   
 
u09524072.png (72)
 
u09524072.png (72)
Line 1,043: Line 1,857:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 141.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|143.]]) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954106.png || $\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$ || $$\empty$$ || conf 0.287
+
| 141.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|143.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954106.png
 +
|| $\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$  
 +
||$$\{g\in \operatorname {GL}(V):(1-g)^n=0\},\quad n =\operatorname {dim}V,$$
 +
|| conf 0.287
 
   
 
   
 
u0954106.png (6)
 
u0954106.png (6)
Line 1,057: Line 1,875:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 142.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|51.]]) ||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090122.png || $\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$ || $$\empty$$ || conf 0.507
+
| 142.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|51.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090122.png
 +
|| $\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$  
 +
||$$\operatorname {diag}(t_1,\ldots ,t_n)\mapsto t _ 1^{\lambda _ 1}\ldots t _ n^{\lambda _ n}\in K,$$
 +
|| conf 0.507
 
   
 
   
 
w120090122.png (122)
 
w120090122.png (122)
 
|-
 
|-
| 143.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|54.]])*||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090135.png || $\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ || $$\empty$$ || conf 0.461  F  
+
| 143.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|54.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090135.png
 +
|| $\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$  
 +
||$$\chi _ {\lambda }=\sum _ {\mu \in \Lambda (n)}\operatorname {dim}_K(\Delta (\lambda )^{\mu })_{e_{\mu }},$$
 +
|| conf 0.461  F  
 
   
 
   
 
w120090135.png (135)
 
w120090135.png (135)
 
|-
 
|-
| 144.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|110.]]) ||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090259.png || $\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$ || $$\empty$$ || conf 0.381
+
| 144.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|110.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090259.png
 +
|| $\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$  
 +
||$$\mathfrak B=\{e_{\pm }\alpha ,h_{\beta }:\alpha \in \Phi ^{+},\beta \in \Sigma \}.$$
 +
|| conf 0.381
 
   
 
   
 
w120090259.png (259)
 
w120090259.png (259)
 
|-
 
|-
| 145.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|82.]]) ||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090342.png || $\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ || $$\empty$$ || conf 0.487
+
| 145.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|82.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090342.png
 +
|| $\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$  
 +
||$$\left(
 +
\begin {array}ch\\
 +
i
 +
\end {array}
 +
\right)=\frac {h(h-1)\ldots (h-i+1)}{i!}
 +
$$
 +
|| conf 0.487
 
   
 
   
 
w120090342.png (342)
 
w120090342.png (342)
 
|-
 
|-
| 146.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|28.]])*||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009095.png || $\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$ || $$\empty$$ || conf 0.312  F  
+
| 146.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|28.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009095.png
 +
|| $\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$  
 +
||$$\mathfrak S_{\{1,\ldots ,\lambda _ 1\}}\times \mathfrak S_{\{\lambda _ 1+1,\ldots ,\lambda _ 1+\lambda _ 2\}}\times \dots $$
 +
|| conf 0.312  F  
 
   
 
   
 
w12009095.png (95)
 
w12009095.png (95)
 
|-
 
|-
| 147.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|104.]]) ||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009096.png || $\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ || $$\empty$$ || conf 0.259
+
| 147.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|104.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009096.png
 +
|| $\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$  
 +
||$$\ldots \times \mathfrak S_{\{\lambda _ 1+\ldots +\lambda _ {n-1}+1,\ldots ,r\}},$$
 +
|| conf 0.259
 
   
 
   
 
w12009096.png (96)
 
w12009096.png (96)
Line 1,091: Line 1,938:
 
!style=width: 3%| Nr.
 
!style=width: 3%| Nr.
 
!style=width: 30%| Image of png File
 
!style=width: 30%| Image of png File
!style=width: 30%| $\TeX$, 1st version
+
!style=width: 30%| $\TeX$, automatically generated version
!style=width: 30%| $\TeX$, corrected version
+
!style=width: 30%| $\TeX$, manually corrected version
 
!style=width: 7%| Confidence, F?
 
!style=width: 7%| Confidence, F?
 
png file  
 
png file  
 
|-
 
|-
| 148.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|87.]])*||  https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100172.png || $\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ || $$\empty$$ || conf 0.351  F  
+
| 148.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|87.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100172.png
 +
|| $\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$  
 +
||$$\langle \alpha ><b\rangle =\langle \alpha b \rangle ,\quad \langle {\bf 1}\rangle ={\bf f}_1={\bf V}_1=\text{ unit element}1,$$
 +
|| conf 0.351  F  
 
   
 
   
 
w098100172.png (172)
 
w098100172.png (172)
 
|-
 
|-
| 149.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|123.]])*||  https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100177.png || $\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ || $$\empty$$ || conf 0.143  F  
+
| 149.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|123.]])*
 +
||  https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100177.png
 +
|| $\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$  
 +
||$$\langle \alpha +b\rangle =\sum _ {n=1}^{\infty }{\bf V}_n\langle r _ n(\alpha ,b){\bf f}_n.$$
 +
|| conf 0.143  F  
 
   
 
   
 
w098100177.png (177)
 
w098100177.png (177)
 
|-
 
|-
| 150.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|102.]]) ||  https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100190.png || $\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$ || $$\empty$$ || conf 0.771
+
| 150.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|102.]])  
 +
||  https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100190.png
 +
|| $\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$  
 +
||$$\sigma (\alpha _ 1,\alpha _ 2,\ldots )=(\alpha _ 1^p,\alpha _ 2^p,\ldots )$$
 +
|| conf 0.771
 
   
 
   
 
w098100190.png (190)
 
w098100190.png (190)
 
|-
 
|-
 
|}
 
|}

Latest revision as of 17:27, 11 November 2019

This page gives an analysis of the code here, generated automatically from some png files underlying our old wiki pages. As this page does contain a lot of $\TeX$ code, it loads slowly.

Under the name of some of our EoM-pages the table below lists some png files, displaying their image and their $TeX$ rendering (automatically retrieved and corrected by hand). The first column gives the running number in this table, followed (in parentheses) by the number used here. The last column gives the confidence and the name of the png file, followed (in parentheses) by the number it has in the sequence of all png files called by its calling EoM-page.

Here is a short survey of the more systematic errors which seem to occur:

1. Trailing punctuation is dismissed.
[concerns almost all images] ; technically: pixels in sparse last pixel columns of bit images are suppressed/ignored?
2. "Displayed" images are not recognized as such.
[concerns almost all images]
Therefore these are displayed too small, and like "inline" $\TeX$ format.
Remark: This cannot be discovered from the png file, it has to be retrieved from the html markup in the calling file: Displayed images are embedded in some html <table> markup.
3. Sparse initial column pixels of the bit image are dismissed
(in parts this affects essential symbols), [see nr. 15,16,36,43,58,59,60,61,62,63,97,109]
4. Some fonts are not recognized
\cal: [7.12.25.26,30,31,32,33,95,111] \mathbf: [30,83,111,127] \bf:[ 133,148,149]
5. Semi-colon is interpreted as double pipe = "||"
[33,49,86,101]
6. Some code is not displayed at all.
(This seems to be a bug of our MathJax TeX interpreter.) [67,74,78,81,83,94,101,106]
This seems to happen when a string "\text {" is involved, can apparently be fixed by using "\text{", but still unclear.
7. Questions
The different interpretation of the matrix delimiters in [56-63] is a bit surprising. Should be checked!
Also, the vanishing of some '-' signs in the first column of some matrices, maybe that is related to 3.?

Algebraic curve

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

1.(23.) a01145065.png $g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$ $$g\leq \left\{ \begin {array}{ll} {\frac {(n-2)^2}4} &{\text{ for even }n,}\\ {\frac {(n-1)(n-3)}4} &{\text{ for odd }n,} \end {array} \right.$$ conf 0.698

a01145065.png (65)

Algebraic geometry

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

2.(116.) a01150014.png $\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ $$\theta =\int\limits _ 0^{\lambda }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ conf 0.997

a01150014.png (14)

3.(133.) a01150021.png $\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ $$\omega =2\int\limits _ 0^{1/c}\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ conf 0.973

a01150021.png (21)

4.(67.) a01150022.png $\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ $$\widetilde w=2\int\limits _ 0^{1/\varepsilon }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ conf 0.107

a01150022.png (22)

5.(105.) a01150044.png $\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ $$\theta (v+\pi i r )=\theta (r),\quad \theta (v+\alpha _ j)=e^{L_j(v)}\theta (v),$$ conf 0.775

a01150044.png (44)

6.(17.) a01150078.png $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ $$\left( \begin {array}{ll} {\alpha } &b\\ c &d \end {array} \right)\equiv \left( \begin {array}{ll} 1&0\\ 0&1 \end {array} \right)(\operatorname {mod}7).$$ conf 0.440

a01150078.png (78)

Algebraic surface

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

7.(144.) a011640132.png $0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$ $$0\rightarrow {\cal O}_V\rightarrow E _ {\alpha }\rightarrow T _ V\rightarrow 0$$ conf 0.981

a011640132.png (132)

8.(73.) a011640137.png $M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$ $$M=\operatorname {dim}\operatorname {Im}(H^1(V,E_{\alpha })\rightarrow H ^1(V,T_V)).$$ conf 0.997

a011640137.png (137)

9.(88.) a011640139.png $\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$ $$\operatorname {dim}_kH^2(V,E_{\alpha })+\operatorname {dim}_kH^2(V,T_V).$$ conf 0.996

a011640139.png (139)

10.(117.) a01164027.png $N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$ $$N_m=\left(\begin {array}c{m+3}\\ 3 \end {array} \right)-dm+2t+\tau +p-1.$$ conf 0.369

a01164027.png (27)

11.(72.) a01164029.png $p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$ $$p_{\alpha }(V)=\left(\begin {array}c{n-1}\\ 3 \end {array} \right)-d(n-1)+2t+\tau +p-1$$ conf 0.396

a01164029.png (29)

12.(68.)* a01164047.png $p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$ $$p_{\alpha }(V)=-\operatorname {dim}_kH_1(V,{\cal O}_V)+\operatorname {dim}_kH^2(V,{\cal O}_V)=$$ conf 0.756 F

a01164047.png (47)

13.(93.)* a01164053.png $1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$ $$1+p_{\alpha }(V)=\frac {\operatorname {deg}(c_1^2)+\operatorname {deg}(c_2)}{12},$$ conf 0.752 F

a01164053.png (53)

Cartan subalgebra

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

14.(33.)* c0205509.png $\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$ $$\mathfrak g_0=\big\{X\in \mathfrak g:\forall H \in \mathfrak t\exists n_{X,H}\in {\mathbb Z}((\text{ ad }H)^{n_{X,H}}(X)=0)\big\},$$ conf 0.110 F

c0205509.png (9)

Cartan theorem

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

15.(49.)* c0205704.png $f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ $$[e_i,f_j]=\delta _ {ij}h_i,\quad [h_i,e_j]=\alpha _ {ij}e_j,\quad [h_i,f_j]=-\alpha _ {ij}f_j,$$ conf 0.149 F

c0205704.png (4)

16.(55.)* c02057064.png $\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$ $$\dots \rightarrow H ^p(X,S)\rightarrow H ^p(X,F)\stackrel {\phi_p }{\rightarrow }H^p(X,G)\rightarrow $$ conf 0.853 F

c02057064.png (64)

Comitant

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

17.(7.) c02333033.png $H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ $$H=\frac 1{36}\left| \begin {array}{cc} {\frac {\partial ^2f}{\partial x ^2}} &{\frac {\partial ^2f}{\partial x \partial y }}\\ {\frac {\partial ^2f}{\partial x \partial y }} &{\frac {\partial ^2f}{\partial y ^2}} \end {array} \right|=$$ conf 0.956

c02333033.png (33)

18.(76.) c02333034.png $= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$ $$=(a_0a_2-a_1^2)x^2+(a_0a_3-a_1a_2)xy+(a_1a_3-a_2^2)y^2$$ conf 0.549

c02333034.png (34)

19.(11.)* c02333035.png $( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ $$(\alpha _ 0,\alpha _ 1,\alpha _ 2,\alpha _ 3)\mapsto (\alpha _ 0\alpha _ 2-\alpha _ 1^2,\frac 12(\alpha _ 0\alpha _ 3-\alpha _ 1\alpha _ 2),\alpha _ 1\alpha _ 3-\alpha _ 2^2)$$ conf 0.521 F

c02333035.png (35)

Deformation

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

20.(26.) d030700175.png $\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ $$\operatorname {Aut}_{R^{\prime }}(X^{\prime }|X_0)\rightarrow \operatorname {Aut}_R(X_{R^{\prime }}^{\prime }\otimes R |X_0)$$ conf 0.683
\

d030700175.png (175)

21.(27.) d030700190.png $\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$ $$\operatorname {dim}_kH^1(X_0,T_{X_0})-\operatorname {dim}M_{X_0}\leq \operatorname {dim}_kH^2(X_0,T_{X_0}).$$ conf 0.944

d030700190.png (190)

22.(78.)* d030700263.png $\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$ $$\alpha \circ b =\alpha b +\sum _ {i=1}^{\infty }\phi _ i(\alpha ,b)t^i,\quad \alpha ,b\in V,$$ conf 0.097 F

d030700263.png (263)

23.(96.)* d030700270.png $\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$ $$\Phi (\alpha )=\alpha +\sum _ {i=1}^{\infty }t^i\phi _ i(\alpha ),\quad \alpha \in V,$$ conf 0.873 F

d030700270.png (270)

Differential algebra

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

24.(106.) d031830107.png $S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ $$S^tF=\sum _ {j=1}^rc_jA^{p_j}A_1^{i_{1j}}\dots A _ {m-l}^{i_{{m-l},j}},$$ conf 0.149

d031830107.png (107)

25.(146.)* d031830141.png $( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$ $(\eta _ 1,\ldots ,\eta _ k)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ k)$ conf 0.562 F

d031830141.png (141)

26.(145.)$^F$* d031830150.png $( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$ $(\eta _ 1,\ldots ,\eta _ n)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ n)$ conf 0.376 F

d031830150.png (150)

27.(57.) d03183016.png $\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ $$\omega _ V=\sum _ {0\leq i \leq m }\alpha _ i\left( \begin {array}c{x+i}\\ i \end {array} \right),$$ conf 0.780

d03183016.png (16)

28.(111.) d03183043.png $e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$ $$e_{ij}=\operatorname {ord}_{Y_j}F_i,\quad 1 \leq i \leq n ,\quad i \leq j \leq n,$$ conf 0.187

d03183043.png (43)

Dimension polynomial

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

29.(48.) d03249029.png $\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ $$\omega _ {\eta /F}(x)=\sum _ {0\leq i \leq m }\alpha _ i\left(\begin {array}c{x+i}\\ i \end {array} \right),$$ conf 0.968

d03249029.png (29)

Duality

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

30.(118.)* d034120173.png $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ $$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow {\mathbf C},$$ conf 0.824 F

d034120173.png (173)

31.(59.)* d034120175.png $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$ $$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow H _ c^n(X,\Omega )$$ conf 0.921 F

d034120175.png (175)

32.(124.)* d034120184.png $( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$ $$(H^p(X,{\cal F}))^{\prime }\cong H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega )).$$ conf 0.829 F

d034120184.png (184)

33.(29.)* d034120236.png $\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X F , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y || F , \Omega )$ $$\beta :\operatorname {Ext}_c^{n-p-1}(X;{\cal F},\Omega )\rightarrow \operatorname {Ext}_c^{n-p-1}(X\backslash Y ;{\cal F},\Omega ).$$ conf 0.634 F

d034120236.png (236)

34.(77.)* d034120247.png $\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$ $$\underset {n\rightarrow \infty }{\overline {\lim }}|\alpha _ n|^{1/n}=\sigma <+\infty.$$ conf 0.521 F

d034120247.png (247)

35.(58.)* d034120253.png $h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$ $$h(\phi )=\underset {n\rightarrow \infty }{\overline {\lim }}\frac {\operatorname {ln}|A(re^{i\phi })|}r$$ conf 0.861 F

d034120253.png (253)

36.(69.)* d034120360.png $\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$ $$\operatorname*{sup}_{l\in E^\perp \atop \|l\|\le 1 }|l(\omega )|=\operatorname*{inf}_{x\in E }\|\omega -x\|,$$ conf 0.293 F

d034120360.png (360)

37.(15.) d034120376.png $\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$ $$\operatorname*{sup}_{f\in B ^1}\big|\int\limits _ {\partial G }f(\zeta )\omega (\zeta )d\zeta \big|=\operatorname*{inf}_{\phi \in E ^1}\int\limits _ {\partial G }|\omega (\zeta )-\phi (\zeta ) ||d\zeta |.$$ conf 0.508

d034120376.png (376)

38.(52.) d034120509.png $f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ $$f=\{f_{\alpha }\}\in \prod _ {\alpha }F_{\alpha },\quad g =\{g_{\alpha }\}\in \operatorname*\oplus _ {\alpha }G_{\alpha }.$$ conf 0.491

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39.(140.) d034120535.png $f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$ $$f^{*}(x^{*})=\operatorname*{sup}_{x\in X }(\langle x ^{*},x\rangle -f(x))$$ conf 0.900

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40.(94.) d034120555.png $f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$ $$f_0(x)\rightarrow \text{ inf, }\quad f _ i(x)\leq 0 ,\quad i =1,\ldots ,m,\quad x \in B,$$ conf 0.810

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41.(74.)* d03412079.png $( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ $$(c_{\gamma },c^r)=\sum _ {t^r\in K }c_r(t^{\prime })c^r(t^r)\operatorname {mod}1$$ conf 0.117 F

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Extension of a differential field

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42.(63.) e03696024.png $F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ $$F_1F_2=F_1\langle F _ 2\rangle =F_1(F_2)=F_2(F_1)=F_2\langle F _ 1\rangle,$$ conf 0.628

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Formal group

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43.(120.)* f040820118.png $\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ $$\operatorname {log}F_{\rm MU }(X)=\sum _ {i=1}^{\infty }i^{-1}[{\rm CP}^{i-1}]X^i,$$ conf 0.098 F

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44.(147.)* f04082059.png $( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ $$(x_1,\ldots ,x_n)\circ (y_1,\ldots ,y_n)=(z_1,\ldots ,z_n),$$ conf 0.553 F

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Gel'fond-Schneider method

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45.(148.) g1300205.png $\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$ $\alpha ^{\beta }=\operatorname {exp}\{\beta \operatorname {log}\alpha \}$ conf 0.979

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Group

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46.(22.)* g04521075.png $\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$ source incomplete conf 0.226 F

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Homogeneous space

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47.(89.) h04769069.png $\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$ $$\mathfrak g=\mathfrak f+\mathfrak m,\quad \mathfrak f\cap \mathfrak m=\{0\},$$ conf 0.793

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Hopf algebra

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48.(103.) h047970129.png $m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$ $m\circ (\iota \otimes 1 )\circ \mu =m\circ (1\otimes \iota )\circ \mu =e\circ \epsilon$ conf 0.618

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49.(107.)* h047970139.png $F _ { 1 } ( X || Y ) , \ldots , F _ { n } ( X || Y ) \in K [ X _ { 1 } , \ldots , X _ { n } || Y _ { 1 } , \ldots , Y _ { n } ] \}$ $F_1(X;Y),\ldots ,F_n(X;Y)\in K [X_1,\ldots ,X_n;Y_1,\ldots ,Y_n]\}$ conf 0.353 F

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50.(97.) h04797042.png $\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$ $$\epsilon (x)=0,\quad \delta (x)=x\otimes 1 +1\otimes x ,\quad x \in \mathfrak g.$$ conf 0.213

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Invariants, theory of

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51.(149.)* i05235015.png $\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ $$\alpha _ {i_1,\dots,i_n}\rightarrow \alpha _ {i_1,\dots,i_n}^{\prime }.$$ conf 0.142 F

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Jordan algebra

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52.(150.) j05427030.png $H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ $$(C_3,\Gamma )=\big\{X\in C _ 3:X=\Gamma ^{-1}X\square ^{\prime }\Gamma \big\},$$ conf 0.651

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53.(42.) j05427031.png $\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$ $$\Gamma =\operatorname {diag}\{\gamma _ 1,\gamma _ 2,\gamma _ 3\},\quad \gamma _ i\neq 0 ,\quad \gamma _ i\in F,$$ conf 0.987

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54.(125.)* j05427077.png $\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$ $\mathfrak g=\mathfrak g_{-1}+\mathfrak g_0+\mathfrak g_1$ conf 0.598 F

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Jordan matrix

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55.(6.)* j0543403.png $J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$ $$J=\left\| \begin {array}{cccc} J_{n_1}(\lambda_1) &0 &0 &0\\ 0 &\ddots &\ddots &0\\ 0 &\ddots &\ddots &0\\ 0 &0 &0 &J_{n_s}(\lambda_s) \end {array} \right\|,$$ conf 0.072 F

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56.(64.) j05434030.png $C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ $$C_m(\lambda )=\operatorname {rk}(A-\lambda E )^{m-1}-2\operatorname {rk}(A-\lambda E )^m+$$ conf 0.955

j05434030.png (30)

57.(1.)* j0543406.png $J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$ $$J_m(\lambda)=\left\| \begin {array}{cccccc} \lambda &1 &\square &\square &\square &\square \\ \square &\lambda &1 &\square &0 &\square \\ \square &\square &\ddots &\ddots &\square &\square\\ \square &\square &\square &\ddots &\ddots &\square \\ \square &0 &\square &\square &\lambda &1\\ \square &\square &\square &\square &\square &\lambda \end {array} \right\|,$$ conf 0.098 F

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Lie algebra, semi-simple

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58.(5.) l058510127.png $\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ $$B_n:\quad \left\| \begin {array}{rrrrrr} 2 &{-1} &0 &{\dots } &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot \\ 0 &0 &0 &{\dots } &{-1} &0\\ 0 &0 &0 &{\dots } &2 &{-2}\\ 0 &0 &0 &{\dots } &{-1} &2 \end {array} \right\|,$$ conf 0.232

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59.(3.)* l058510129.png $\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ $$D_n:\quad \left\| \begin {array}{rrrrrrr} 2 &{-1} &0 &{\dots } &0 &0 &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0 &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0 &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot &\cdot &\cdot \\ 0 &0 &0 &{\dots } &2 &{-1} &0 &0\\ 0 &0 &0 &{\dots } &{-1} &2 &{-1} &{-1}\\ 0 &0 &0 &{\dots } &0 &{-1} &2 &0\\ 0 &0 &0 &{\dots } &0 &{-1} &0 &2 \end {array} \right\|,$$ conf 0.055 F

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60.(8.)* l058510130.png $\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ $$E_6: \quad \left\| \begin {array}{rrrrrr} 2 &0 &{-1} &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0\\ {-1} &0 &2 &{-1} &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0\\ 0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ conf 0.628 F

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61.(4.) l058510131.png $\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ $$E_7:\quad \left\| \begin {array}{rrrrrrr} 2 &0 &{-1} &0 &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ conf 0.278

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62.(2.)* l058510132.png $\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ $$E_8:\quad \left\| \begin {array}{rrrrrrrr} 2 &0 &{-1} &0 &0 &0 &0 & 0\\ 0 &2 &0 &{-1} &0 &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ conf 0.354 F

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63.(10.)* l058510133.png $\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$ $$F_4:\quad \left\| \begin {array}{rrrr} 2 &{-1} &0 &0\\ {-1} &2 &{-2} &0\\ 0 &{-1} &2 &{-1}\\ 0 &0 &{-1} &2 \end {array} \right\|,\quad G _ 2:\quad \left\| \begin {array}{rr} 2&{-1}\\ {-3}&2 \end {array} \right\|.$$ conf 0.374 F

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64.(98.) l05851030.png $\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$ $$\mathfrak g_{\alpha }=\{X\in \mathfrak g:[H,X]=\alpha (H)X,H\in \mathfrak h\}.$$ conf 0.976

l05851030.png (30)

65.(126.) l05851037.png $\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ $$\mathfrak g=\mathfrak h+\sum _ {\alpha \in \Sigma }\mathfrak g_{\alpha }.$$ conf 0.945

l05851037.png (37)

66.(61.)* l05851044.png $\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$ $$\mathfrak g_{\alpha }=\operatorname {dim}[\mathfrak g_{\alpha },\mathfrak g_{-\alpha }]=1.$$ conf 0.520 F

l05851044.png (44)

67.(65.)* l05851050.png $[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ $$[H_{\alpha },X_{\alpha }]=2X_{\alpha }\quad {\rm and }\quad [H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }.$$ conf 0.539 F

l05851050.png (50)

68.(70.) l05851051.png $\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$ $$\beta (H_{\alpha })=\frac {2(\alpha ,\beta )}{(\alpha ,\alpha )},\quad \alpha ,\beta \in \Sigma,$$ conf 0.997

l05851051.png (51)

69.(112.) l05851057.png $[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ $$[\mathfrak g_{\alpha },\mathfrak g_{\beta }]=\mathfrak g_{\alpha +\beta }$$ conf 0.917

l05851057.png (57)

70.(127.) l05851064.png $H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$ $$H_{\alpha _ 1},\ldots ,H_{\alpha _ k},X_{\alpha }\quad (\alpha \in \Sigma )$$ conf 0.432

l05851064.png (64)

71.(113.)* l05851069.png $[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ $$[[X_{\alpha _ i},X_{-}\alpha _ i],X_{-\alpha _ j}]=-n(i,j)X_{\alpha _ j},$$ conf 0.628 F

l05851069.png (69)

72.(79.) l05851073.png $n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$ $$n(i,j)=\alpha _ j(H_i)=\frac {2(\alpha _ i,\alpha _ j)}{(\alpha _ j,\alpha _ j)}.$$ conf 0.992

l05851073.png (73)

73.(13.) l05851074.png $[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ $$[X_{\alpha },X_{\beta }]=\left\{ \begin {array}{ll} {N_{\alpha ,\beta }X_{\alpha +\beta }} &{\text{ if }\alpha +\beta \in \Sigma,}\\ 0 &{\text{ if }\alpha +\beta \notin \Sigma,} \end {array} \right.$$ conf 0.988

l05851074.png (74)

74.(80.) l05851078.png $N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$ $$N_{\alpha ,\beta }=-N_{-\alpha ,-\beta }\quad {\rm and }\quad N _ {\alpha ,\beta }=\pm (p+1),$$ conf 0.961

l05851078.png (78)

75.(85.)* l05851085.png $X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ $$iH_\alpha,X_{\alpha }-X_{-\alpha },\quad i (X_{\alpha }+X_{-\alpha })\quad (\alpha \in \Sigma _ {+})$$ conf 0.691 F

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Lie algebra, solvable

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76.(119.)* l05852011.png $[ \mathfrak { g } _ { i } , \mathfrak { g } _ { i } ] \subset \mathfrak { g } _ { \mathfrak { i } } + 1$ $[\mathfrak g_i,\mathfrak g_i]\subset \mathfrak g_{i+1}$ conf 0.276 F

l05852011.png (11)

77.(141.) l05852046.png $\operatorname { dim } \mathfrak { g } _ { i } = \operatorname { dim } \mathfrak { g } - i$ $\operatorname {dim}\mathfrak g_i=\operatorname {dim}\mathfrak g-i$ conf 0.901

l05852046.png (46)

Lie group

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78.(62.)* l058590115.png $( G ) \cong \operatorname { Aut } ( L ( G ) ) \quad \text { and } \quad L ( \operatorname { Aut } ( G ) ) \cong D ( L ( G ) )$ $$\operatorname {Aut}(G)\cong \operatorname {Aut}(L(G))\quad {\rm and }\quad L (\operatorname {Aut}(G))\cong D (L(G)),$$ conf 0.693 F

l058590115.png (115)

79.(50.) l05859086.png $( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ $$(X,Y)\rightarrow \operatorname {exp}^{-1}(\operatorname {exp}X\operatorname {exp}Y),\quad X ,Y\in L (G),$$ conf 0.856

l05859086.png (86)

Lie group, compact

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80.(121.)* l05861012.png $J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$ $$J=\left\| \begin {array}{cc} 0 &{E_x}\\ {-E_x} &0 \end {array} \right\|,$$ conf 0.364 F

l05861012.png (12)

Lie group, nilpotent

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81.(83.) l0586604.png $N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$ $$N(F)=\{g\in GL (V):gv\equiv v \operatorname {mod}V_i\;\text{for all }v\in V _ i,\;i\geq 1 \}$$ conf 0.466

l0586604.png (4)

Lie group, semi-simple

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82.(35.)* l058680102.png $L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$ $$L(\mathfrak g)\cong \Gamma _ 0(\mathfrak u)\cap \mathfrak h^{\prime }/\Gamma _ 0([\mathfrak k,\mathfrak k])$$ conf 0.659 F

l058680102.png (102)

83.(81.)* l05868032.png $\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ $$\Gamma _ 1=\Gamma _ 1(g)=\{X\in h :\alpha (X)\in 2 \pi i {\mathbf Z}\;\text{for all }\alpha \in \Sigma \}.$$ conf 0.183 F

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Lie p-algebra

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84.(36.) l05872026.png $( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$ $$(\operatorname {ad}x)^ny=\sum _ {j=1}^n(-1)^j\left(\begin {array}cn\\ j \end {array} \right)x^{n-j}yx^j$$ conf 0.356

l05872026.png (26)

85.(99.) l05872078.png $\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ $$\pi (x+y)=\pi (x)+\pi (y),\quad \pi (\lambda x )=\lambda ^p\pi (x),\quad \lambda \in k .$$ conf 0.964

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Lie theorem

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86.(134.) l05876010.png $y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ $$y_i=f_i(g_1,\ldots ,g_i;x_1,\ldots ,x_n),\quad i =1,\ldots ,n$$ conf 0.276

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87.(86.) l05876016.png $X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ $$X_i=\sum _ {j=1}^n\xi _ {ij}(x)\frac {\partial }{\partial x _ j},\quad i =1,\ldots ,r,$$ conf 0.656

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88.(66.)* l05876030.png $\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$ $$\frac {\partial f _ j}{\partial g _ i}(g,x)=\sum _ {k=1}^r\xi _ {kj}(f(g_sx))\psi _ {ki}(g),$$ conf 0.336 F

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89.(19.)* l05876037.png $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ $$\sum _ {k=1}^N(\xi _ {ik}\frac {\partial \xi _ {jl}}{\partial x _ k}-\xi _ {jk}\frac {\partial \xi _ {il}}{\partial x _ k})=\sum _ {k=1}^rc_{ij}^k\xi _ {kl},$$ conf 0.157 F

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90.(14.) l05876052.png $\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ $$\left.\begin {array}c{c_{ij}^k=-c_{ji}^k},\\ {\displaystyle\sum _ {l=1}^r(c_{il}^mc_{jk}^l+c_{kl}^mc_{ij}^l+c_{jl}^mc_{ki}^l)=0,\quad 1 \leq i ,j,k,l,m\leq r,} \end {array} \right\}$$ conf 0.085

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Maximal torus

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91.(95.) m06301072.png $F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$ $$F(x_1f_1+\ldots +x_xf_n)=x_1x_n+x_2x_{n-1}+\ldots +x_px_{n-p+1},$$ conf 0.198

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Non-Abelian cohomology

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92.(114.)* n066900110.png $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ $$\phi (g_1)\phi (g_2)\phi (g_1g_2)^{-1}=\operatorname {Int}m(g_1,g_2),$$ conf 0.443 F

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93.(90.)* n066900118.png $( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$ $$m'(g_1,g_2)=h(g_1)(\phi (g_1)(h(g_2)))m(g_1,g_2)h(g_1,g_2)^{-1}.$$ conf 0.764 F

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94.(44.) n06690016.png $\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ $$\delta (e)=e\quad \;\text{and }\quad \delta (\rho (a)b)=\sigma (a)\delta (b),\quad \alpha \in C ^0,\quad b \in C ^1,$$ conf 0.400

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95.(60.)* n06690028.png $C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$ $$C^{*}(\mathfrak U,{\cal F})=(C^0(\mathfrak U,{\cal F}),C^1(\mathfrak U,{\cal F}),C^2(\mathfrak U,{\cal F})),$$ conf 0.205 F

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Picard scheme

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96.(39.)* p07267025.png $\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$ $$\operatorname {Pic}_{X/k}(S^{\prime })=\operatorname {Pic}(X\times_k S ^{\prime })/\operatorname {Pic}(S^{\prime })$$ conf 0.345 F +

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Principal analytic fibration

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97.(100.)* p07464025.png $g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$ $$g_j:U_i\cap U _ j\rightarrow G ,\quad i ,j\in I ,\quad U _ i\cap U _ j\neq \emptyset,$$ conf 0.184 F

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Quantum groups

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98.(101.) q07631062.png $\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$ $$\phi ^{*}:\mathfrak g^{*}\otimes \mathfrak g^{*}\rightarrow \mathfrak g^{*}$$ conf 0.837

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99.(108.) q07631071.png $\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ $$\delta :U_{\mathfrak g}\rightarrow U _ {\mathfrak g}\otimes U _ {\mathfrak g}$$ conf 0.648

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100.(56.)* q07631072.png $\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ $$\delta (\alpha )=\operatorname {lim}_{h\rightarrow 0 }h^{-1}(\Delta (a)-\Delta ^{\prime }(\alpha ))$$ conf 0.304 F

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101.(129.)* q07631088.png $[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$ $$[\alpha ,X_i^{\pm }]=\pm \alpha _ i(\alpha )X_i^{\pm }\quad \text{for }a\in \mathfrak h;$$ conf 0.544 F

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102.(128.) q07631089.png $[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ $$[X_i^{+},X_j^{-}]=2\delta _ {ij}h^{-1}\operatorname {sinh}(hH_i/2).$$ conf 0.893

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103.(20.) q07631092.png $\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$ $$\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)q^{-k(n-k)/2}(X_i^{\pm })^kX_j^{\pm }\cdot (X_i^{\pm })^{n-k}=0.$$ conf 0.055

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104.(30.) q07631095.png $\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ $$\left( \begin {array}ln\\ k \end {array} \right)_q=\frac {(q^n-1)\ldots (q^{n-k+1}-1)}{(q^k-1)\ldots (q-1)} .$$ conf 0.443

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105.(21.)* q07631099.png $\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$ $$\Delta (X_i^{\pm })=X_i^{\pm }\otimes \operatorname {exp}(\frac {hH_i}4)+\operatorname {exp}(\frac {-hH_i}4)\otimes X _ i^{\pm }.$$ conf 0.212 F

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Rational representation

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106.(91.) r077630100.png $0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$ $$0\leq \frac {2(\chi ,\alpha )}{(\alpha ,\alpha )}<p\quad \text{for all }\alpha \in \Delta.$$ conf 0.879

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107.(135.) r077630104.png $\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ $$\phi _ 0\otimes \phi _ 1^{Fr}\otimes \ldots \otimes \phi _ d^{{Fr}^d},$$ conf 0.136

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108.(45.)* r07763055.png $\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$ $$\chi =\delta _ {\phi }-\sum _ {\alpha \in \Delta }m_{\alpha }\alpha ,\quad m _ {\alpha }\in Z ,\quad m _ {\alpha }\geq 0.$$ conf 0.862 F

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Singular point

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109.(31.) s085590225.png $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ $$\sum _ {k_1,\ldots ,k_n=0}^{\infty }c_{k_1\cdots k _ n}(z_1-\zeta _ 1)^{k_1}\ldots (z_n-\zeta _ n)^{k_n}$$ conf 0.324

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110.(46.) s085590404.png $\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ $$\frac {m_1}{n_1}<\frac {m_2}{n_1n_2}<\ldots <\frac {m_g}{n_1\ldots n _ g}=\frac {m_g}n$$ conf 0.459

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111.(115.)* s085590429.png $p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ $$p(Z)=1-\operatorname {dim}H^0({\mathbf Z},{\cal O}_{\mathbf Z })+\operatorname {dim}H^1({\mathbf Z},{\cal O}_{\mathbf Z })$$ conf 0.997 F

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112.(136.)* s085590440.png $X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$ $$X_{\epsilon }=\{(x_0,\ldots ,x_x):f(x_0,\ldots ,x_x)=\epsilon \}$$ conf 0.433 F

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113.(12.) s085590458.png $= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ $$=\left\{ \begin {array}{ll} {(x+\lambda )^2\ldots (x+k\lambda )^2} &{\text{ if }\mu =2k,}\\ {(x+\lambda )^2\ldots (x+k\lambda )^2(x+(k+1)\lambda )} &{\text{ if }\mu =2k+1,} \end {array} \right.$$ conf 0.870

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114.(75.) s085590482.png $( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ $$\big(\frac {\partial F (x,y,\lambda )}{\partial x },\frac {\partial F (x,y,\lambda )}{\partial y }\big)$$ conf 0.986

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115.(137.) s085590515.png $\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$ $$\frac {dx_i}{dx_{i_0}}=f_i(x),\quad f _ i\in C (U),\quad i \neq i _ 0.$$ conf 0.594

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116.(142.)* s085590527.png $A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ $$A=\left\| \begin {array}{ll} {\alpha } &b\\ c &e \end {array} \right\|$$ conf 0.506 F

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117.(53.) s085590634.png $\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$ $$\Delta =(F_{xx}^{\prime \prime })_0(F_{yy}^{\prime \prime })_0-(F_{xy}^{\prime \prime })_0^2$$ conf 0.920

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118.(16.)* s085590645.png $\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ $$\left\| \begin {array}{lll} {F_x^{\prime }} &{F_y^{\prime }} &{F_z^{\prime }}\\ {G_x^{\prime }} &{G_y^{\prime }} &{G_Z^{\prime }} \end {array} \right\|$$ conf 0.230 F

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119.(92.) s085590653.png $( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$ $$(F_x^{\prime })_0=0,\quad (F_y^{\prime })_0=0,\quad (F_z^{\prime })_0=0.$$ conf 0.300

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Solv manifold

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120.(138.) s08610054.png $\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$ $$\{e\}\rightarrow \Delta \rightarrow \pi \rightarrow {\mathbf Z}^s\rightarrow \{e\}$$ conf 0.972

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Stability theorems in algebraic K-theory

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121.(71.) s08706033.png $\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$ $$\psi _ {t_1,\ldots ,t_n}^{\prime }:SK_1(R)\rightarrow S K _ 1(R(t_1,\ldots ,t_n)).$$ conf 0.379

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Steinberg module

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122.(130.) s13053016.png $e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$ $$e=\frac {|U|}{|G|}\big(\sum _ {b\in B }b\big)\big(\sum _ {w\in W }\operatorname {sign}(w)w\big)$$ conf 0.138

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Steinberg symbol

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123.(24.)* s13054017.png $( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ $$(x_{ij}(a),x_{kl}(b))=\left\{ \begin {array}{ll} 1 &{\text{ if }i\neq l ,j\neq k },\\ {x_{il}(ab)} &{\text{ if }i\neq l ,j=k}. \end {array} \right.$$ conf 0.381 F

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Tilting theory

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124.(84.) t130130105.png $0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$ $$0\rightarrow \Lambda \rightarrow T _ 1\rightarrow \ldots \rightarrow T _ n\rightarrow 0 $$ conf 0.946

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Tits quadratic form

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125.(18.) t130140104.png $q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ $$q_R(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}x_ix_j+\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}r_{i,j}x_ix_j,$$ conf 0.112

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126.(40.) t130140118.png $[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$ $$[X]\mapsto \chi _ R([X])=\sum _ {m=0}^{\infty }(-1)^m\operatorname {dim}_K\operatorname {Ext}_R^m(X,X)$$ conf 0.116

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127.(132.)* t130140119.png $\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ $$\underline {\dim }:K_0(\operatorname {mod}R)\rightarrow {\mathbf Z}^{Q_0}$$ conf 0.287 F

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128.(37.)* t130140140.png $q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ $$q_I(x)=\sum _ {i\in I }x_i^2+\sum _ {i\prec j \atop j\in I\setminus {\rm max}I}x_ix_j-\sum _ {p\in \operatorname {max}I}\big(\sum _ {i\prec p }x_i\big)x_p$$ conf 0.197 F

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129.(131.)* t13014044.png $X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ $$X\mapsto \underline {\dim }X=(\operatorname {dim}_KX_j)_{j\in Q _ 0}$$ conf 0.819 F

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130.(25. t13014048.png $[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$ $$[X]\mapsto \chi _ Q([X])=\operatorname {dim}_K\operatorname {End}_Q(X)-\operatorname {dim}_K\operatorname {Ext}_Q^1(X,X)$$ conf 0.661

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131.(38.)* t13014056.png $A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ $$A_Q(v)=\prod _ {i,j\in Q _ 0}\prod _ {\langle \beta :j\rightarrow i \rangle \in Q _ 1}M_{v_i\times v _ j}(K)_{\beta }$$ conf 0.481 F

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132.(139.)* t1301406.png $\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ $$q_Q(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {i,j\in Q _ 0}d_{ij}x_ix_j,$$ conf 0.648 F

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Torus

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133.(41.)* t0933502.png $r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ $$r=\alpha \operatorname {sin}u{\bf k}+l(1+\epsilon \operatorname {cos}u)({\bf i}\operatorname {cos}v+{\bf j}\operatorname {sin}v)$$ conf 0.585 F

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134.(122.)* t0933507.png $d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$ $$ds^2=\alpha ^2du^2+l^2(1+\epsilon \operatorname {cos}u)^2dv^2,$$ conf 0.696 F

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Uniform distribution

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135.(9.) u09524027.png $u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ $$u_3(x)=\left\{ \begin {array}{ll} {\frac {x^2}2,} &{0\leq x <1,}\\ {\frac {[x^2-3(x-1)^2]}2,} &{1\leq x <2,}\\ {\frac {[x^2-3(x-1)^2+3(x-2)^2]}2,} &{2\leq x <3,}\\ {0,} &{x\notin [0,3].} \end {array} \right.$$ conf 0.733

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136.(32.)* u0952403.png $p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$ $$p(x)=\left\{ \begin {array}{ll} {\frac 1{b-\alpha },} &{x\in [\alpha ,b],}\\ {0,} &{x\notin [\alpha ,b].} \end {array} \right.$$ conf 0.681 F

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137.(34.) u09524030.png $u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ $$u_n(x)=\frac 1{(n-1)!}\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)(x-k)_{+}^{n-1}$$ conf 0.569

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138.(109.) u09524034.png $z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$ $$z_{+}=\left\{ \begin {array}{ll} {z,} &{z>0}.\\ {0,} &{z\leq 0 }. \end {array} \right.$$ conf 0.676

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139.(43.) u0952407.png $F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ $$F(x)=\left\{ \begin {array}{ll} {0,} &{x\leq a },\\ {\frac {x-a}{b-a},} &{a<x\leq b },\\ {1,} &{x>b}, \end {array} \right.$$ conf 0.468

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140.(47.) u09524072.png $p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$ $$p(x_1,\ldots ,x_n)=\left\{ \begin {array}{ll} {C\neq 0 ,} &{x\in D },\\ {0,} &{x\notin D }, \end {array} \right.$$ conf 0.705

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Unipotent group

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141.(143.) u0954106.png $\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$ $$\{g\in \operatorname {GL}(V):(1-g)^n=0\},\quad n =\operatorname {dim}V,$$ conf 0.287

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Weyl module

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

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142.(51.) w120090122.png $\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$ $$\operatorname {diag}(t_1,\ldots ,t_n)\mapsto t _ 1^{\lambda _ 1}\ldots t _ n^{\lambda _ n}\in K,$$ conf 0.507

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143.(54.)* w120090135.png $\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ $$\chi _ {\lambda }=\sum _ {\mu \in \Lambda (n)}\operatorname {dim}_K(\Delta (\lambda )^{\mu })_{e_{\mu }},$$ conf 0.461 F

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144.(110.) w120090259.png $\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$ $$\mathfrak B=\{e_{\pm }\alpha ,h_{\beta }:\alpha \in \Phi ^{+},\beta \in \Sigma \}.$$ conf 0.381

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145.(82.) w120090342.png $\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ $$\left( \begin {array}ch\\ i \end {array} \right)=\frac {h(h-1)\ldots (h-i+1)}{i!} $$ conf 0.487

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146.(28.)* w12009095.png $\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$ $$\mathfrak S_{\{1,\ldots ,\lambda _ 1\}}\times \mathfrak S_{\{\lambda _ 1+1,\ldots ,\lambda _ 1+\lambda _ 2\}}\times \dots $$ conf 0.312 F

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147.(104.) w12009096.png $\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ $$\ldots \times \mathfrak S_{\{\lambda _ 1+\ldots +\lambda _ {n-1}+1,\ldots ,r\}},$$ conf 0.259

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Witt vector

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

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148.(87.)* w098100172.png $\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ $$\langle \alpha ><b\rangle =\langle \alpha b \rangle ,\quad \langle {\bf 1}\rangle ={\bf f}_1={\bf V}_1=\text{ unit element}1,$$ conf 0.351 F

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149.(123.)* w098100177.png $\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ $$\langle \alpha +b\rangle =\sum _ {n=1}^{\infty }{\bf V}_n\langle r _ n(\alpha ,b){\bf f}_n.$$ conf 0.143 F

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150.(102.) w098100190.png $\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$ $$\sigma (\alpha _ 1,\alpha _ 2,\ldots )=(\alpha _ 1^p,\alpha _ 2^p,\ldots )$$ conf 0.771

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How to Cite This Entry:
Ulf Rehmann/Table of automatically generated TeX code. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ulf_Rehmann/Table_of_automatically_generated_TeX_code&oldid=44168