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− | This page gives an analysis of [[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|the code here]], generated automatically from some png files underlying our old wiki pages. | + | 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 | + | 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]]== | ||
Line 7: | Line 37: | ||
!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%| $\TeX$, | + | !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 || $g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$ || | + | | 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 21: | 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%| $\TeX$, | + | !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 || $\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || | + | | 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 || $\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || | + | | 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 || $\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || $$\widetilde | + | | 4.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|67.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150022.png | ||
+ | || $\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ | ||
+ | ||$$\widetilde w=2\int\limits _ 0^{1/\varepsilon }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ | ||
+ | || conf 0.107 | ||
a01150022.png (22) | a01150022.png (22) | ||
|- | |- | ||
− | | 5.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|105.]]) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150044.png || $\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ || | + | | 5.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|105.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150044.png | ||
+ | || $\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ | ||
+ | ||$$\theta (v+\pi i r )=\theta (r),\quad \theta (v+\alpha _ j)=e^{L_j(v)}\theta (v),$$ | ||
+ | || conf 0.775 | ||
a01150044.png (44) | a01150044.png (44) | ||
|- | |- | ||
− | | 6.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|17.]]) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150078.png || $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ || | + | | 6.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|17.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150078.png | ||
+ | || $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ | ||
+ | ||$$\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).$$ | ||
+ | || conf 0.440 | ||
a01150078.png (78) | a01150078.png (78) | ||
Line 52: | Line 121: | ||
!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 7.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|144.]]) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640132.png || $0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$ || $$0 \rightarrow {\cal O} | + | | 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 || $M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$ || $$ M = \operatorname { dim } \operatorname { Im } ( H ^ | + | | 8.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|73.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640137.png | ||
+ | || $M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$ | ||
+ | ||$$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 } | + | | 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 || $N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$ || $$ | + | | 10.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|117.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164027.png | ||
+ | || $N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$ | ||
+ | ||$$N_m=\left(\begin {array}c{m+3}\\ | ||
+ | 3 | ||
+ | \end {array} | ||
+ | \right)-dm+2t+\tau +p-1.$$ | ||
+ | || conf 0.369 | ||
a01164027.png (27) | a01164027.png (27) | ||
|- | |- | ||
− | | 11.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|72.]]) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164029.png || $p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$ || $$ | + | | 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 || $p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$ || | + | | 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 || $1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$ || | + | | 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 91: | 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%| $\TeX$, | + | !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 || $\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$ || | + | | 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) | ||
Line 106: | Line 213: | ||
!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%| $\TeX$, | + | !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 || $f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ || $$ [e_i, | + | | 15.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|49.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205704.png | ||
+ | || $f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ | ||
+ | ||$$[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,$$ | ||
+ | || conf 0.149 F | ||
c0205704.png (4) | c0205704.png (4) | ||
|- | |- | ||
− | | 16.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|55.]])*|| https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057064.png || $\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$ || $$ \dots \rightarrow H ^ | + | | 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) | ||
Line 125: | Line 240: | ||
!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%| $\TeX$, | + | !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 || $H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ || $$H = \frac | + | | 17.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|7.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333033.png | ||
+ | || $H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ | ||
+ | ||$$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|=$$ | ||
+ | || conf 0.956 | ||
c02333033.png (33) | c02333033.png (33) | ||
|- | |- | ||
− | | 18.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|76.]]) || https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333034.png || $= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$ || $$ = ( | + | | 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 || $( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ || $$ ( \alpha _ | + | | 19.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|11.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333035.png | ||
+ | || $( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ | ||
+ | ||$$(\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)$$ | ||
+ | || conf 0.521 F | ||
c02333035.png (35) | c02333035.png (35) | ||
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!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 20.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|26.]]) || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700175.png || $\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ || | + | | 20.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|26.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700175.png | ||
+ | || $\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ | ||
+ | ||$$\operatorname {Aut}_{R^{\prime }}(X^{\prime }|X_0)\rightarrow \operatorname {Aut}_R(X_{R^{\prime }}^{\prime }\otimes R |X_0)$$ | ||
+ | || conf 0.683 | ||
\ | \ | ||
d030700175.png (175) | d030700175.png (175) | ||
|- | |- | ||
− | | 21.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|27.]]) || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700190.png || $\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$ || $$ \operatorname { dim } | + | | 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 || $\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$ || $$ \alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ | + | | 22.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|78.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700263.png | ||
+ | || $\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$ | ||
+ | ||$$\alpha \circ b =\alpha b +\sum _ {i=1}^{\infty }\phi _ i(\alpha ,b)t^i,\quad \alpha ,b\in V,$$ | ||
+ | || conf 0.097 F | ||
d030700263.png (263) | d030700263.png (263) | ||
|- | |- | ||
− | | 23.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|96.]])*|| https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700270.png || $\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$ || $$ \Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ | + | | 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) | ||
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− | !style=width: 30%| $\TeX$, | + | !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 || $S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ || $$ S ^ | + | | 24.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|106.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830107.png | ||
+ | || $S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ | ||
+ | ||$$S^tF=\sum _ {j=1}^rc_jA^{p_j}A_1^{i_{1j}}\dots A _ {m-l}^{i_{{m-l},j}},$$ | ||
+ | || conf 0.149 | ||
d031830107.png (107) | d031830107.png (107) | ||
|- | |- | ||
− | | 25.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|146.]])*|| https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830141.png || $( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$ || $ ( \eta _ | + | | 25.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|146.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830141.png | ||
+ | || $( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$ | ||
+ | ||$(\eta _ 1,\ldots ,\eta _ k)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ k)$ | ||
+ | || conf 0.562 F | ||
d031830141.png (141) | d031830141.png (141) | ||
|- | |- | ||
− | | 26.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|145.]])$^F$*|| https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830150.png || $( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$ || $ ( \eta _ | + | | 26.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|145.]])$^F$* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830150.png | ||
+ | || $( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$ | ||
+ | ||$(\eta _ 1,\ldots ,\eta _ n)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ n)$ | ||
+ | || conf 0.376 F | ||
d031830150.png (150) | d031830150.png (150) | ||
|- | |- | ||
− | | 27.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|57.]]) || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d03183016.png || $\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ || $$ \omega _ | + | | 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 || $e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$ || | + | | 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) | ||
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!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%| $\TeX$, | + | !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 || $\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ || | + | | 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) | ||
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− | !style=width: 30%| $\TeX$, | + | !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 || $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ || $$ H ^ | + | | 30.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|118.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120173.png | ||
+ | || $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ | ||
+ | ||$$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow {\mathbf C},$$ | ||
+ | || conf 0.824 F | ||
d034120173.png (173) | d034120173.png (173) | ||
|- | |- | ||
− | | 31.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|59.]])*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120175.png || $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$ || $$ H ^ | + | | 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 || $( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$ || $$ ( H ^ | + | | 32.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|124.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120184.png | ||
+ | || $( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$ | ||
+ | ||$$(H^p(X,{\cal F}))^{\prime }\cong H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega )).$$ | ||
+ | || conf 0.829 F | ||
d034120184.png (184) | d034120184.png (184) | ||
|- | |- | ||
− | | 33.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|29.]])*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120236.png || $\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X F , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y || F , \Omega )$ || $$ \beta : \operatorname { Ext } | + | | 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 || $\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$ || $$ \underset { n \rightarrow \infty } { \overline { \lim } } | \alpha _ | + | | 34.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|77.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120247.png | ||
+ | || $\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$ | ||
+ | ||$$\underset {n\rightarrow \infty }{\overline {\lim }}|\alpha _ n|^{1/n}=\sigma <+\infty.$$ | ||
+ | || conf 0.521 F | ||
d034120247.png (247) | d034120247.png (247) | ||
|- | |- | ||
− | | 35.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|58.]])*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120253.png || $h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$ || $$ h ( \phi ) = \underset { n\rightarrow \infty }{\overline{ \lim } } \frac { \operatorname { ln } | A ( | + | | 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 || $\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$ || $$ \operatorname* { sup } _ { l \in E^\perp \atop \|l\|\le 1 } | + | | 36.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|69.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120360.png | ||
+ | || $\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$ | ||
+ | ||$$\operatorname*{sup}_{l\in E^\perp \atop \|l\|\le 1 }|l(\omega )|=\operatorname*{inf}_{x\in E }\|\omega -x\|,$$ | ||
+ | || conf 0.293 F | ||
d034120360.png (360) | d034120360.png (360) | ||
|- | |- | ||
− | | 37.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|15.]]) || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120376.png || $\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$ || $$ \operatorname* { sup } _ { f \in B ^ | + | | 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 || $f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ || $$ f = \{ | + | | 38.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|52.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120509.png | ||
+ | || $f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ | ||
+ | ||$$f=\{f_{\alpha }\}\in \prod _ {\alpha }F_{\alpha },\quad g =\{g_{\alpha }\}\in \operatorname*\oplus _ {\alpha }G_{\alpha }.$$ | ||
+ | || conf 0.491 | ||
d034120509.png (509) | d034120509.png (509) | ||
|- | |- | ||
− | | 39.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|140.]]) || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120535.png || $f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$ || $$ f ^ { * } ( x ^ { * } ) = \operatorname*{ sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) ) $$|| conf 0.900 | + | | 39.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|140.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120535.png | ||
+ | || $f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$ | ||
+ | ||$$f^{*}(x^{*})=\operatorname*{sup}_{x\in X }(\langle x ^{*},x\rangle -f(x))$$ | ||
+ | || conf 0.900 | ||
d034120535.png (535) | d034120535.png (535) | ||
|- | |- | ||
− | | 40.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|94.]]) || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120555.png || $f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$ || $$ | + | | 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 || $( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ || $$ ( | + | | 41.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|74.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d03412079.png | ||
+ | || $( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ | ||
+ | ||$$(c_{\gamma },c^r)=\sum _ {t^r\in K }c_r(t^{\prime })c^r(t^r)\operatorname {mod}1$$ | ||
+ | || conf 0.117 F | ||
d03412079.png (79) | d03412079.png (79) | ||
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− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 42.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|63.]]) || https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696024.png || $F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ || | + | | 42.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|63.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696024.png | ||
+ | || $F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ | ||
+ | ||$$F_1F_2=F_1\langle F _ 2\rangle =F_1(F_2)=F_2(F_1)=F_2\langle F _ 1\rangle,$$ | ||
+ | || conf 0.628 | ||
e03696024.png (24) | e03696024.png (24) | ||
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!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 43.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|120.]])*|| https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f040820118.png || $\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ || | + | | 43.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|120.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f040820118.png | ||
+ | || $\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ | ||
+ | ||$$\operatorname {log}F_{\rm MU }(X)=\sum _ {i=1}^{\infty }i^{-1}[{\rm CP}^{i-1}]X^i,$$ | ||
+ | || conf 0.098 F | ||
f040820118.png (118) | f040820118.png (118) | ||
|- | |- | ||
− | | 44.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|147.]])*|| https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f04082059.png || $( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ || | + | | 44.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|147.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f04082059.png | ||
+ | || $( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ | ||
+ | ||$$(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) | ||
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− | !style=width: 30%| $\TeX$, | + | !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 \}$ || | + | | 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) | ||
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− | !style=width: 30%| $\TeX$, | + | !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.$ | + | | 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) | ||
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!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$, | + | !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 \}$ || | + | | 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) | ||
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!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
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− | !style=width: 30%| $\TeX$, | + | !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$ || $m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$ | + | | 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 } ] \}$ || $ | + | | 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 }$ || | + | | 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.$$ | ||
+ | || conf 0.213 | ||
h04797042.png (42) | h04797042.png (42) | ||
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− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 51.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|149.]])*|| https://www.encyclopediaofmath.org/legacyimages/i/i052/i052350/i05235015.png || $\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ || | + | | 51.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|149.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/i/i052/i052350/i05235015.png | ||
+ | || $\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ | ||
+ | ||$$\alpha _ {i_1,\dots,i_n}\rightarrow \alpha _ {i_1,\dots,i_n}^{\prime }.$$ | ||
+ | || conf 0.142 F | ||
i05235015.png (15) | i05235015.png (15) | ||
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− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 52.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|150.]]) || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427030.png || $H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ || | + | | 52.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|150.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427030.png | ||
+ | || $H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ | ||
+ | ||$$(C_3,\Gamma )=\big\{X\in C _ 3:X=\Gamma ^{-1}X\square ^{\prime }\Gamma \big\},$$ | ||
+ | || conf 0.651 | ||
j05427030.png (30) | j05427030.png (30) | ||
|- | |- | ||
− | | 53.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|42.]]) || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427031.png || $\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$ || | + | | 53.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|42.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427031.png | ||
+ | || $\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$ | ||
+ | ||$$\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 }$ || $\mathfrak | + | | 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) | ||
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− | !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|$ || | + | | 55.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|6.]])* |
− | J_{n_1}(\lambda_1) | + | || 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}{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 || $C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ || $$ | + | | 56.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|64.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434030.png | ||
+ | || $C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ | ||
+ | ||$$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} { | + | | 57.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|1.]])* |
− | \lambda & | + | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543406.png |
− | \square & \lambda & | + | || $J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$ |
− | \square & \square & \ddots | + | ||$$J_m(\lambda)=\left\| |
− | \square & \square & \square & | + | \begin {array}{cccccc} |
− | \square & | + | |
− | \square & \square & \square & | + | \lambda &1 &\square &\square &\square &\square \\ |
− | \end{array} \right\|,$$ || 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\|,$$ | ||
+ | || conf 0.098 F | ||
j0543406.png (6) | j0543406.png (6) | ||
Line 454: | Line 769: | ||
!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 58.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|5.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510127.png || $\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ || $$B_n: \quad \left\| \begin{array} { | + | | 58.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|5.]]) |
− | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510127.png | |
− | { - 1 } & | + | || $\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ |
− | + | ||$$B_n:\quad \left\| | |
− | \cdot | + | \begin {array}{rrrrrr} |
− | + | ||
− | + | 2 &{-1} &0 &{\dots } &0 &0\\ | |
− | + | ||
− | \end{array} \right\|,$$|| 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\|,$$ | ||
+ | || conf 0.232 | ||
l058510127.png (127) | l058510127.png (127) | ||
|- | |- | ||
− | | 59.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|3.]])*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510129.png || $\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ || | + | | 59.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|3.]])* |
− | \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510129.png |
− | + | || $\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ | |
− | { - 1 } & | + | ||$$D_n:\quad \left\| |
− | + | \begin {array}{rrrrrrr} | |
− | \cdot | + | |
− | + | 2 &{-1} &0 &{\dots } &0 &0 &0 &0\\ | |
− | + | ||
− | + | {-1} &2 &{-1} &{\dots } &0 &0 &0 &0\\ | |
− | + | ||
− | \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 || $\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ || $$E_6: | + | | 60.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|8.]])* |
− | \quad \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510130.png |
− | + | || $\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ | |
− | + | ||$$E_6: | |
− | { - 1 } & | + | \quad \left\| |
− | + | \begin {array}{rrrrrr} | |
− | + | ||
− | + | 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 || $\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ || $$E_7: \quad | + | | 61.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|4.]]) |
− | \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510131.png |
− | + | || $\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ | |
− | + | ||$$E_7:\quad \left\| | |
− | {-1 } & | + | \begin {array}{rrrrrrr} |
− | + | ||
− | + | 2 &0 &{-1} &0 &0 &0 &0\\ | |
− | + | ||
− | + | 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 || $\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ || $$E_8: \quad | + | | 62.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|2.]])* |
− | \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510132.png |
− | + | || $\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ | |
− | + | ||$$E_8:\quad \left\| | |
− | {-1 } & | + | \begin {array}{rrrrrrrr} |
− | + | ||
− | + | 2 &0 &{-1} &0 &0 &0 &0 & | |
− | + | 0\\ | |
− | + | 0 &2 &0 &{-1} &0 &0 &0 &0\\ | |
− | + | ||
− | \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.]])* |
− | || $$F_4: \quad \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510133.png |
+ | || $\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$ | ||
+ | |||
+ | ||$$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\|.$$ | ||
+ | || conf 0.374 F | ||
l058510133.png (133) | l058510133.png (133) | ||
|- | |- | ||
− | | 64.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|98.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851030.png || $\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$ || $$ \mathfrak | + | | 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 || $\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ || $$ \mathfrak | + | | 65.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|126.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851037.png | ||
+ | || $\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ | ||
+ | ||$$\mathfrak g=\mathfrak h+\sum _ {\alpha \in \Sigma }\mathfrak g_{\alpha }.$$ | ||
+ | || conf 0.945 | ||
l05851037.png (37) | l05851037.png (37) | ||
|- | |- | ||
− | | 66.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|61.]])*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851044.png || $\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$ || $$ \mathfrak | + | | 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 || $[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ || $$ [ | + | | 67.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|65.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851050.png | ||
+ | || $[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ | ||
+ | ||$$[H_{\alpha },X_{\alpha }]=2X_{\alpha }\quad {\rm and }\quad [H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }.$$ | ||
+ | || conf 0.539 F | ||
l05851050.png (50) | l05851050.png (50) | ||
|- | |- | ||
− | | 68.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|70.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851051.png || $\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$ || $$ \beta ( | + | | 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 || $[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ || $$ [ \mathfrak | + | | 69.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|112.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851057.png | ||
+ | || $[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ | ||
+ | ||$$[\mathfrak g_{\alpha },\mathfrak g_{\beta }]=\mathfrak g_{\alpha +\beta }$$ | ||
+ | || conf 0.917 | ||
l05851057.png (57) | l05851057.png (57) | ||
|- | |- | ||
− | | 70.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|127.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851064.png || $H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$ || $$ | + | | 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 || $[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ || $$ [ [ | + | | 71.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|113.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851069.png | ||
+ | || $[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ | ||
+ | ||$$[[X_{\alpha _ i},X_{-}\alpha _ i],X_{-\alpha _ j}]=-n(i,j)X_{\alpha _ j},$$ | ||
+ | || conf 0.628 F | ||
l05851069.png (69) | l05851069.png (69) | ||
|- | |- | ||
− | | 72.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|79.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851073.png || $n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$ || $$ n ( i , j ) = \alpha _ | + | | 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 || $[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ || $$ [ | + | | 73.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|13.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851074.png | ||
+ | || $[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ | ||
+ | ||$$[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.$$ | ||
+ | || conf 0.988 | ||
l05851074.png (74) | l05851074.png (74) | ||
|- | |- | ||
− | | 74.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|80.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851078.png || $N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$ || $$ | + | | 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 || $X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ || $$ iH_\alpha, | + | | 75.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|85.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851085.png | ||
+ | || $X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ | ||
+ | ||$$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) | ||
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!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !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$ || | + | | 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$ || | + | | 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) | ||
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− | !style=width: 30%| $\TeX$, | + | !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 ) )$ || | + | | 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)),$$ | ||
+ | || conf 0.693 F | ||
l058590115.png (115) | l058590115.png (115) | ||
|- | |- | ||
− | | 79.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|50.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l05859086.png || $( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ || | + | | 79.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|50.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l05859086.png | ||
+ | || $( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ | ||
+ | ||$$(X,Y)\rightarrow \operatorname {exp}^{-1}(\operatorname {exp}X\operatorname {exp}Y),\quad X ,Y\in L (G),$$ | ||
+ | || conf 0.856 | ||
l05859086.png (86) | l05859086.png (86) | ||
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!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 80.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|121.]])*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058610/l05861012.png || $J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$ || | + | | 80.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|121.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058610/l05861012.png | ||
+ | || $J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$ | ||
+ | ||$$J=\left\| | ||
+ | \begin {array}{cc} | ||
+ | 0 &{E_x}\\ | ||
+ | {-E_x} &0 | ||
+ | \end {array} | ||
+ | \right\|,$$ | ||
+ | || conf 0.364 F | ||
l05861012.png (12) | l05861012.png (12) | ||
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!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 81.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|83.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586604.png || $N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$ || | + | | 81.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|83.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586604.png | ||
+ | || $N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$ | ||
+ | ||$$N(F)=\{g\in GL (V):gv\equiv v \operatorname {mod}V_i\;\text{for all }v\in V _ i,\;i\geq 1 \}$$ | ||
+ | || conf 0.466 | ||
l0586604.png (4) | l0586604.png (4) | ||
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− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 82.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|35.]])*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680102.png || $L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$ || $$ L ( \mathfrak | + | | 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 || $\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ || $$ \Gamma _ | + | | 83.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|81.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868032.png | ||
+ | || $\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ | ||
+ | ||$$\Gamma _ 1=\Gamma _ 1(g)=\{X\in h :\alpha (X)\in 2 \pi i {\mathbf Z}\;\text{for all }\alpha \in \Sigma \}.$$ | ||
+ | || conf 0.183 F | ||
l05868032.png (32) | l05868032.png (32) | ||
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− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 84.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|36.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872026.png || $( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$ || $$ ( \operatorname { ad } x ) ^ | + | | 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 || $\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ || $$ \pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ | + | | 85.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|99.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872078.png | ||
+ | || $\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ | ||
+ | ||$$\pi (x+y)=\pi (x)+\pi (y),\quad \pi (\lambda x )=\lambda ^p\pi (x),\quad \lambda \in k .$$ | ||
+ | || conf 0.964 | ||
l05872078.png (78) | l05872078.png (78) | ||
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!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 86.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|134.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876010.png || $y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ || $$ | + | | 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 || $X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ || $$ | + | | 87.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|86.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876016.png | ||
+ | || $X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ | ||
+ | ||$$X_i=\sum _ {j=1}^n\xi _ {ij}(x)\frac {\partial }{\partial x _ j},\quad i =1,\ldots ,r,$$ | ||
+ | || conf 0.656 | ||
l05876016.png (16) | l05876016.png (16) | ||
|- | |- | ||
− | | 88.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|66.]])*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876030.png || $\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$ || $$ \frac { \partial 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 || $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ || $$ \sum _ { k = 1 } ^ | + | | 89.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|19.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876037.png | ||
+ | || $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ | ||
+ | ||$$\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},$$ | ||
+ | || conf 0.157 F | ||
l05876037.png (37) | l05876037.png (37) | ||
|- | |- | ||
− | | 90.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|14.]]) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876052.png || $\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ || $$ \left. \begin{array} | + | | 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) | ||
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− | !style=width: 30%| $\TeX$, | + | !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 || $F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$ || | + | | 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) | ||
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!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !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 || $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ || $$ \phi ( | + | | 92.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|114.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900110.png | ||
+ | || $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ | ||
+ | ||$$\phi (g_1)\phi (g_2)\phi (g_1g_2)^{-1}=\operatorname {Int}m(g_1,g_2),$$ | ||
+ | || conf 0.443 F | ||
n066900110.png (110) | n066900110.png (110) | ||
|- | |- | ||
− | | 93.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|90.]])*|| https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900118.png || $( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$ || $$ m'( | + | | 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 || $\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ || $$ \delta ( e ) = e \quad \;\text {and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ | + | | 94.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|44.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690016.png | ||
+ | || $\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ | ||
+ | ||$$\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,$$ | ||
+ | || conf 0.400 | ||
n06690016.png (16) | n06690016.png (16) | ||
|- | |- | ||
− | | 95.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|60.]])*|| https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690028.png || $C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$ || $$ C ^ { * } ( \mathfrak | + | | 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) | ||
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− | !style=width: 30%| $\TeX$, | + | !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 || $\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$ || $$ \operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Pic } ( X \times_k S ^ { \prime } ) / \operatorname { Pic } ( S ^ { \prime } ) $$|| 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%| $\TeX$, | + | !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 || $g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$ || $$ g_j : | + | | 97.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|100.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464025.png | ||
+ | || $g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$ | ||
+ | ||$$g_j:U_i\cap U _ j\rightarrow G ,\quad i ,j\in I ,\quad U _ i\cap U _ j\neq \emptyset,$$ | ||
+ | || conf 0.184 F | ||
p07464025.png (25) | p07464025.png (25) | ||
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− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 98.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|101.]]) || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631062.png || $\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$ || $$ \phi ^ { * } : \mathfrak | + | | 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 || $\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ || $$ \delta : | + | | 99.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|108.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631071.png | ||
+ | || $\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ | ||
+ | ||$$\delta :U_{\mathfrak g}\rightarrow U _ {\mathfrak g}\otimes U _ {\mathfrak g}$$ | ||
+ | || conf 0.648 | ||
q07631071.png (71) | q07631071.png (71) | ||
|- | |- | ||
− | | 100.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|56.]])*|| https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631072.png || $\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ || $$ \delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) ) $$|| conf 0.304 F | + | | 100.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|56.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631072.png | ||
+ | || $\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ | ||
+ | ||$$\delta (\alpha )=\operatorname {lim}_{h\rightarrow 0 }h^{-1}(\Delta (a)-\Delta ^{\prime }(\alpha ))$$ | ||
+ | || conf 0.304 F | ||
q07631072.png (72) | q07631072.png (72) | ||
|- | |- | ||
− | | 101.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|129.]])*|| https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631088.png || $[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$ || $$ [ \alpha , | + | | 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 || $[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ || $$ [ | + | | 102.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|128.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631089.png | ||
+ | || $[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ | ||
+ | ||$$[X_i^{+},X_j^{-}]=2\delta _ {ij}h^{-1}\operatorname {sinh}(hH_i/2).$$ | ||
+ | || conf 0.893 | ||
q07631089.png (89) | q07631089.png (89) | ||
|- | |- | ||
− | | 103.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|20.]]) || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631092.png || $\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$ || $$ \sum _ { k = 0 } ^ | + | | 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 || $\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ || $$ \left( \begin{array} | + | | 104.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|30.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631095.png | ||
+ | || $\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ | ||
+ | ||$$\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)} | ||
+ | .$$ | ||
+ | || conf 0.443 | ||
q07631095.png (95) | q07631095.png (95) | ||
|- | |- | ||
− | | 105.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|21.]])*|| https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631099.png || $\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$ || $$ \Delta ( | + | | 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) | ||
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− | !style=width: 30%| $\TeX$, | + | !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 || $0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$ || $$ 0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text {for all } \alpha \in \Delta. $$|| 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 || $\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ || $$ \phi _ | + | | 107.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|135.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630104.png | ||
+ | || $\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ | ||
+ | ||$$\phi _ 0\otimes \phi _ 1^{Fr}\otimes \ldots \otimes \phi _ d^{{Fr}^d},$$ | ||
+ | || conf 0.136 | ||
r077630104.png (104) | r077630104.png (104) | ||
|- | |- | ||
− | | 108.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|45.]])*|| https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763055.png || $\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$ || $$ \chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } | + | | 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) | ||
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− | !style=width: 30%| $\TeX$, | + | !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 | ||
|- | |- | ||
− | | 109.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|31.]]) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590225.png || $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ || $$ \sum _ { | + | | 109.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|31.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590225.png | ||
+ | || $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ | ||
+ | ||$$\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}$$ | ||
+ | || conf 0.324 | ||
s085590225.png (225) | s085590225.png (225) | ||
|- | |- | ||
− | | 110.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|46.]]) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590404.png || $\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ || $$ \frac { | + | | 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 || $p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ || $$ p ( Z ) = 1 - \operatorname { dim } H ^ | + | | 111.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|115.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590429.png | ||
+ | || $p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ | ||
+ | ||$$p(Z)=1-\operatorname {dim}H^0({\mathbf Z},{\cal O}_{\mathbf Z })+\operatorname {dim}H^1({\mathbf Z},{\cal O}_{\mathbf Z })$$ | ||
+ | || conf 0.997 F | ||
s085590429.png (429) | s085590429.png (429) | ||
|- | |- | ||
− | | 112.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|136.]])*|| https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590440.png || $X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$ || $$ | + | | 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 || $= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ || $$ = \left\{ \begin{array} { | + | | 113.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|12.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590458.png | ||
+ | || $= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ | ||
+ | ||$$=\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.$$ | ||
+ | || conf 0.870 | ||
s085590458.png (458) | s085590458.png (458) | ||
|- | |- | ||
− | | 114.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|75.]]) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590482.png || $( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ || $$ \big( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } \big) $$|| conf 0.986 | + | | 114.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|75.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590482.png | ||
+ | || $( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ | ||
+ | ||$$\big(\frac {\partial F (x,y,\lambda )}{\partial x },\frac {\partial F (x,y,\lambda )}{\partial y }\big)$$ | ||
+ | || conf 0.986 | ||
s085590482.png (482) | s085590482.png (482) | ||
|- | |- | ||
− | | 115.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|137.]]) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590515.png || $\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$ || $$ \frac { | + | | 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 || $A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ || $$ A = | + | | 116.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|142.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590527.png | ||
+ | || $A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ | ||
+ | ||$$A=\left\| | ||
+ | \begin {array}{ll} | ||
+ | {\alpha } &b\\ | ||
+ | c &e | ||
+ | \end {array} | ||
+ | \right\|$$ | ||
+ | || conf 0.506 F | ||
s085590527.png (527) | s085590527.png (527) | ||
|- | |- | ||
− | | 117.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|53.]]) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590634.png || $\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$ || $$ \Delta = ( | + | | 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 || $\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ || $$ \left\| \begin{array} { | + | | 118.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|16.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590645.png | ||
+ | || $\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ | ||
+ | ||$$\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\|$$ | ||
+ | || conf 0.230 F | ||
s085590645.png (645) | s085590645.png (645) | ||
|- | |- | ||
− | | 119.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|92.]]) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590653.png || $( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$ || $$ ( | + | | 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) | ||
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!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 || $\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$ || $$ \{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow {\mathbf Z} ^ | + | | 120.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|138.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610054.png | ||
+ | || $\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$ | ||
+ | ||$$\{e\}\rightarrow \Delta \rightarrow \pi \rightarrow {\mathbf Z}^s\rightarrow \{e\}$$ | ||
+ | || conf 0.972 | ||
s08610054.png (54) | s08610054.png (54) | ||
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− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 121.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|71.]]) || https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706033.png || $\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$ || $$ \psi _ { | + | | 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) | ||
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!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !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 || $e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$ || $$ e = \frac { | U | } { | G | } \big( \sum _ { b \in B } b \big) \big( \sum _ { w \in W } \operatorname { sign } ( w ) w \big) $$|| 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) | ||
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!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%| $\TeX$, | + | !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 || $( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ || $$ ( | + | | 123.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|24.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054017.png | ||
+ | || $( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ | ||
+ | ||$$(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.$$ | ||
+ | || conf 0.381 F | ||
s13054017.png (17) | s13054017.png (17) | ||
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!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 124.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|84.]]) || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130105.png || $0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$ || $$ 0 \rightarrow \Lambda \rightarrow T _ | + | | 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) | ||
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!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%| $\TeX$, | + | !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 || $q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ || $$ q_R ( x ) = \sum _ { j \in Q _ | + | | 125.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|18.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140104.png | ||
+ | || $q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ | ||
+ | ||$$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,$$ | ||
+ | || conf 0.112 | ||
t130140104.png (104) | t130140104.png (104) | ||
|- | |- | ||
− | | 126.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|40.]]) || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140118.png || $[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$ || $$ [ X ] \mapsto \chi _ | + | | 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 || $\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ || $$ \ | + | | 127.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|132.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140119.png | ||
+ | || $\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ | ||
+ | ||$$\underline {\dim }:K_0(\operatorname {mod}R)\rightarrow {\mathbf Z}^{Q_0}$$ | ||
+ | || conf 0.287 F | ||
t130140119.png (119) | t130140119.png (119) | ||
|- | |- | ||
− | | 128.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|37.]])*|| https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140140.png || $q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ || $$ q_I ( x ) = \sum _ { i \in I } | + | | 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 || $X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ || $$ X \mapsto \ | + | | 129.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|131.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014044.png | ||
+ | || $X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ | ||
+ | ||$$X\mapsto \underline {\dim }X=(\operatorname {dim}_KX_j)_{j\in Q _ 0}$$ | ||
+ | || conf 0.819 F | ||
t13014044.png (44) | t13014044.png (44) | ||
|- | |- | ||
− | | 130.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|25.]] | + | | 130.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|25.]] |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014048.png | ||
+ | || $[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$ | ||
+ | ||$$[X]\mapsto \chi _ Q([X])=\operatorname {dim}_K\operatorname {End}_Q(X)-\operatorname {dim}_K\operatorname {Ext}_Q^1(X,X)$$ | ||
+ | || conf 0.661 | ||
t13014048.png (48) | t13014048.png (48) | ||
|- | |- | ||
− | | 131.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|38.]])*|| https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014056.png || $A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ || $$ | + | | 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 || $\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ || $$ q_Q ( x ) = \sum _ { j \in Q _ | + | | 132.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|139.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301406.png | ||
+ | || $\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ | ||
+ | ||$$q_Q(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {i,j\in Q _ 0}d_{ij}x_ix_j,$$ | ||
+ | || conf 0.648 F | ||
t1301406.png (6) | t1301406.png (6) | ||
|- | |- | ||
|} | |} | ||
+ | |||
==[[Torus]]== | ==[[Torus]]== | ||
{| class="wikitable" style="text-align: left; width: 1740px;" | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 133.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|41.]])*|| https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933502.png || $r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ || $$ r = \alpha \operatorname { sin } u {\bf k} + l ( 1 + \epsilon \operatorname { cos } u ) ( {\bf i} \operatorname { cos } v + {\bf j} \operatorname { sin } v ) $$|| conf 0.585 F | + | | 133.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|41.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933502.png | ||
+ | || $r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ | ||
+ | ||$$r=\alpha \operatorname {sin}u{\bf k}+l(1+\epsilon \operatorname {cos}u)({\bf i}\operatorname {cos}v+{\bf j}\operatorname {sin}v)$$ | ||
+ | || conf 0.585 F | ||
t0933502.png (2) | t0933502.png (2) | ||
|- | |- | ||
− | | 134.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|122.]])*|| https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933507.png || $d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$ || $$ | + | | 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 1,039: | 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%| $\TeX$, | + | !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 || $u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ || $$ | + | | 135.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|9.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524027.png | ||
+ | || $u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ | ||
+ | ||$$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.$$ | ||
+ | || conf 0.733 | ||
u09524027.png (27) | u09524027.png (27) | ||
|- | |- | ||
− | | 136.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|32.]])*|| https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952403.png || $p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$ || $$ p ( x ) = \left\{ \begin{array} { | + | | 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 || $u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ || $$ | + | | 137.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|34.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524030.png | ||
+ | || $u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ | ||
+ | ||$$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}$$ | ||
+ | || conf 0.569 | ||
u09524030.png (30) | u09524030.png (30) | ||
|- | |- | ||
− | | 138.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|109.]]) || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524034.png || $z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$ || $$ | + | | 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 || $F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ || $$ F ( x ) = \left\{ \begin{array} { | + | | 139.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|43.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952407.png | ||
+ | || $F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ | ||
+ | ||$$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.$$ | ||
+ | || conf 0.468 | ||
u0952407.png (7) | u0952407.png (7) | ||
|- | |- | ||
− | | 140.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|47.]]) || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524072.png || $p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$ || $$ p ( | + | | 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 1,073: | 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%| $\TeX$, | + | !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 || $\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$ || $$ \{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ | + | | 141.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|143.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954106.png | ||
+ | || $\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$ | ||
+ | ||$$\{g\in \operatorname {GL}(V):(1-g)^n=0\},\quad n =\operatorname {dim}V,$$ | ||
+ | || conf 0.287 | ||
u0954106.png (6) | u0954106.png (6) | ||
Line 1,087: | Line 1,875: | ||
!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 142.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|51.]]) || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090122.png || $\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$ || $$ \operatorname { diag } ( | + | | 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 || $\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ || $$ \chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } | + | | 143.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|54.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090135.png | ||
+ | || $\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ | ||
+ | ||$$\chi _ {\lambda }=\sum _ {\mu \in \Lambda (n)}\operatorname {dim}_K(\Delta (\lambda )^{\mu })_{e_{\mu }},$$ | ||
+ | || conf 0.461 F | ||
w120090135.png (135) | w120090135.png (135) | ||
|- | |- | ||
− | | 144.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|110.]]) || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090259.png || $\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$ || $$ \mathfrak | + | | 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 || $\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ || $$ \left( \begin{array} | + | | 145.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|82.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090342.png | ||
+ | || $\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ | ||
+ | ||$$\left( | ||
+ | \begin {array}ch\\ | ||
+ | i | ||
+ | \end {array} | ||
+ | \right)=\frac {h(h-1)\ldots (h-i+1)}{i!} | ||
+ | $$ | ||
+ | || conf 0.487 | ||
w120090342.png (342) | w120090342.png (342) | ||
|- | |- | ||
− | | 146.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|28.]])*|| https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009095.png || $\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$ || $$ \mathfrak | + | | 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 || $\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ || $$ \ldots \times \mathfrak | + | | 147.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|104.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009096.png | ||
+ | || $\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ | ||
+ | ||$$\ldots \times \mathfrak S_{\{\lambda _ 1+\ldots +\lambda _ {n-1}+1,\ldots ,r\}},$$ | ||
+ | || conf 0.259 | ||
w12009096.png (96) | w12009096.png (96) | ||
Line 1,121: | Line 1,938: | ||
!style=width: 3%| Nr. | !style=width: 3%| Nr. | ||
!style=width: 30%| Image of png File | !style=width: 30%| Image of png File | ||
− | !style=width: 30%| $\TeX$, | + | !style=width: 30%| $\TeX$, automatically generated version |
− | !style=width: 30%| $\TeX$, corrected version | + | !style=width: 30%| $\TeX$, manually corrected version |
!style=width: 7%| Confidence, F? | !style=width: 7%| Confidence, F? | ||
png file | png file | ||
|- | |- | ||
− | | 148.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|87.]])*|| https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100172.png || $\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ || $$ \langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle {\bf 1} \rangle = {\bf 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 || $\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ || $$ \langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } {\bf V} | + | | 149.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|123.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100177.png | ||
+ | || $\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ | ||
+ | ||$$\langle \alpha +b\rangle =\sum _ {n=1}^{\infty }{\bf V}_n\langle r _ n(\alpha ,b){\bf f}_n.$$ | ||
+ | || conf 0.143 F | ||
w098100177.png (177) | w098100177.png (177) | ||
|- | |- | ||
− | | 150.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|102.]]) || https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100190.png || $\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$ || $$ \sigma ( \alpha _ | + | | 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.) | ![]() |
$g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$ | $$g\leq \left\{ \begin {array}{ll} {\frac {(n-2)^2}4} &{\text{ for even }n,}\\ {\frac {(n-1)(n-3)}4} &{\text{ for odd }n,} \end {array} \right.$$ | 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.) | ![]() |
$\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ | $$\theta =\int\limits _ 0^{\lambda }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ | conf 0.997
a01150014.png (14) |
3.(133.) | ![]() |
$\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ | $$\omega =2\int\limits _ 0^{1/c}\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ | conf 0.973
a01150021.png (21) |
4.(67.) | ![]() |
$\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ | $$\widetilde w=2\int\limits _ 0^{1/\varepsilon }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ | conf 0.107
a01150022.png (22) |
5.(105.) | ![]() |
$\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ | $$\theta (v+\pi i r )=\theta (r),\quad \theta (v+\alpha _ j)=e^{L_j(v)}\theta (v),$$ | conf 0.775
a01150044.png (44) |
6.(17.) | ![]() |
$\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ | $$\left( \begin {array}{ll} {\alpha } &b\\ c &d \end {array} \right)\equiv \left( \begin {array}{ll} 1&0\\ 0&1 \end {array} \right)(\operatorname {mod}7).$$ | 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.) | ![]() |
$0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$ | $$0\rightarrow {\cal O}_V\rightarrow E _ {\alpha }\rightarrow T _ V\rightarrow 0$$ | conf 0.981
a011640132.png (132) |
8.(73.) | ![]() |
$M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$ | $$M=\operatorname {dim}\operatorname {Im}(H^1(V,E_{\alpha })\rightarrow H ^1(V,T_V)).$$ | conf 0.997
a011640137.png (137) |
9.(88.) | ![]() |
$\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$ | $$\operatorname {dim}_kH^2(V,E_{\alpha })+\operatorname {dim}_kH^2(V,T_V).$$ | conf 0.996
a011640139.png (139) |
10.(117.) | ![]() |
$N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$ | $$N_m=\left(\begin {array}c{m+3}\\ 3 \end {array} \right)-dm+2t+\tau +p-1.$$ | conf 0.369
a01164027.png (27) |
11.(72.) | ![]() |
$p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$ | $$p_{\alpha }(V)=\left(\begin {array}c{n-1}\\ 3 \end {array} \right)-d(n-1)+2t+\tau +p-1$$ | conf 0.396
a01164029.png (29) |
12.(68.)* | ![]() |
$p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$ | $$p_{\alpha }(V)=-\operatorname {dim}_kH_1(V,{\cal O}_V)+\operatorname {dim}_kH^2(V,{\cal O}_V)=$$ | conf 0.756 F
a01164047.png (47) |
13.(93.)* | ![]() |
$1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$ | $$1+p_{\alpha }(V)=\frac {\operatorname {deg}(c_1^2)+\operatorname {deg}(c_2)}{12},$$ | 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.)* | ![]() |
$\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$ | $$\mathfrak g_0=\big\{X\in \mathfrak g:\forall H \in \mathfrak t\exists n_{X,H}\in {\mathbb Z}((\text{ ad }H)^{n_{X,H}}(X)=0)\big\},$$ | 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.)* | ![]() |
$f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ | $$[e_i,f_j]=\delta _ {ij}h_i,\quad [h_i,e_j]=\alpha _ {ij}e_j,\quad [h_i,f_j]=-\alpha _ {ij}f_j,$$ | conf 0.149 F
c0205704.png (4) |
16.(55.)* | ![]() |
$\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$ | $$\dots \rightarrow H ^p(X,S)\rightarrow H ^p(X,F)\stackrel {\phi_p }{\rightarrow }H^p(X,G)\rightarrow $$ | 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 |
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17.(7.) | ![]() |
$H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ | $$H=\frac 1{36}\left| \begin {array}{cc} {\frac {\partial ^2f}{\partial x ^2}} &{\frac {\partial ^2f}{\partial x \partial y }}\\ {\frac {\partial ^2f}{\partial x \partial y }} &{\frac {\partial ^2f}{\partial y ^2}} \end {array} \right|=$$ | conf 0.956
c02333033.png (33) |
18.(76.) | ![]() |
$= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$ | $$=(a_0a_2-a_1^2)x^2+(a_0a_3-a_1a_2)xy+(a_1a_3-a_2^2)y^2$$ | conf 0.549
c02333034.png (34) |
19.(11.)* | ![]() |
$( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ | $$(\alpha _ 0,\alpha _ 1,\alpha _ 2,\alpha _ 3)\mapsto (\alpha _ 0\alpha _ 2-\alpha _ 1^2,\frac 12(\alpha _ 0\alpha _ 3-\alpha _ 1\alpha _ 2),\alpha _ 1\alpha _ 3-\alpha _ 2^2)$$ | conf 0.521 F
c02333035.png (35) |
Deformation
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20.(26.) | ![]() |
$\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ | $$\operatorname {Aut}_{R^{\prime }}(X^{\prime }|X_0)\rightarrow \operatorname {Aut}_R(X_{R^{\prime }}^{\prime }\otimes R |X_0)$$ | conf 0.683
\ d030700175.png (175) |
21.(27.) | ![]() |
$\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$ | $$\operatorname {dim}_kH^1(X_0,T_{X_0})-\operatorname {dim}M_{X_0}\leq \operatorname {dim}_kH^2(X_0,T_{X_0}).$$ | conf 0.944
d030700190.png (190) |
22.(78.)* | ![]() |
$\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$ | $$\alpha \circ b =\alpha b +\sum _ {i=1}^{\infty }\phi _ i(\alpha ,b)t^i,\quad \alpha ,b\in V,$$ | conf 0.097 F
d030700263.png (263) |
23.(96.)* | ![]() |
$\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$ | $$\Phi (\alpha )=\alpha +\sum _ {i=1}^{\infty }t^i\phi _ i(\alpha ),\quad \alpha \in V,$$ | conf 0.873 F
d030700270.png (270) |
Differential algebra
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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24.(106.) | ![]() |
$S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ | $$S^tF=\sum _ {j=1}^rc_jA^{p_j}A_1^{i_{1j}}\dots A _ {m-l}^{i_{{m-l},j}},$$ | conf 0.149
d031830107.png (107) |
25.(146.)* | ![]() |
$( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$ | $(\eta _ 1,\ldots ,\eta _ k)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ k)$ | conf 0.562 F
d031830141.png (141) |
26.(145.)$^F$* | ![]() |
$( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$ | $(\eta _ 1,\ldots ,\eta _ n)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ n)$ | conf 0.376 F
d031830150.png (150) |
27.(57.) | ![]() |
$\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ | $$\omega _ V=\sum _ {0\leq i \leq m }\alpha _ i\left( \begin {array}c{x+i}\\ i \end {array} \right),$$ | conf 0.780
d03183016.png (16) |
28.(111.) | ![]() |
$e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$ | $$e_{ij}=\operatorname {ord}_{Y_j}F_i,\quad 1 \leq i \leq n ,\quad i \leq j \leq n,$$ | conf 0.187
d03183043.png (43) |
Dimension polynomial
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29.(48.) | ![]() |
$\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ | $$\omega _ {\eta /F}(x)=\sum _ {0\leq i \leq m }\alpha _ i\left(\begin {array}c{x+i}\\ i \end {array} \right),$$ | conf 0.968
d03249029.png (29) |
Duality
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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30.(118.)* | ![]() |
$H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ | $$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow {\mathbf C},$$ | conf 0.824 F
d034120173.png (173) | |
31.(59.)* | ![]() |
$H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$ | $$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow H _ c^n(X,\Omega )$$ | conf 0.921 F
d034120175.png (175) | |
32.(124.)* | ![]() |
$( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$ | $$(H^p(X,{\cal F}))^{\prime }\cong H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega )).$$ | conf 0.829 F
d034120184.png (184) | |
33.(29.)* | ![]() |
$\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X F , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y || F , \Omega )$ | $$\beta :\operatorname {Ext}_c^{n-p-1}(X;{\cal F},\Omega )\rightarrow \operatorname {Ext}_c^{n-p-1}(X\backslash Y ;{\cal F},\Omega ).$$ | conf 0.634 | F
d034120236.png (236) |
34.(77.)* | ![]() |
$\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$ | $$\underset {n\rightarrow \infty }{\overline {\lim }}|\alpha _ n|^{1/n}=\sigma <+\infty.$$ | conf 0.521 F
d034120247.png (247) | |
35.(58.)* | ![]() |
$h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$ | $$h(\phi )=\underset {n\rightarrow \infty }{\overline {\lim }}\frac {\operatorname {ln}|A(re^{i\phi })|}r$$ | conf 0.861 F
d034120253.png (253) | |
36.(69.)* | ![]() |
$\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$ | $$\operatorname*{sup}_{l\in E^\perp \atop \|l\|\le 1 }|l(\omega )|=\operatorname*{inf}_{x\in E }\|\omega -x\|,$$ | conf 0.293 F
d034120360.png (360) | |
37.(15.) | ![]() |
$\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$ | $$\operatorname*{sup}_{f\in B ^1}\big|\int\limits _ {\partial G }f(\zeta )\omega (\zeta )d\zeta \big|=\operatorname*{inf}_{\phi \in E ^1}\int\limits _ {\partial G }|\omega (\zeta )-\phi (\zeta ) ||d\zeta |.$$ | conf 0.508
d034120376.png (376) | |
38.(52.) | ![]() |
$f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ | $$f=\{f_{\alpha }\}\in \prod _ {\alpha }F_{\alpha },\quad g =\{g_{\alpha }\}\in \operatorname*\oplus _ {\alpha }G_{\alpha }.$$ | conf 0.491
d034120509.png (509) | |
39.(140.) | ![]() |
$f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$ | $$f^{*}(x^{*})=\operatorname*{sup}_{x\in X }(\langle x ^{*},x\rangle -f(x))$$ | conf 0.900
d034120535.png (535) | |
40.(94.) | ![]() |
$f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$ | $$f_0(x)\rightarrow \text{ inf, }\quad f _ i(x)\leq 0 ,\quad i =1,\ldots ,m,\quad x \in B,$$ | conf 0.810
d034120555.png (555) | |
41.(74.)* | ![]() |
$( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ | $$(c_{\gamma },c^r)=\sum _ {t^r\in K }c_r(t^{\prime })c^r(t^r)\operatorname {mod}1$$ | conf 0.117 F
d03412079.png (79) |
Extension of a differential field
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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42.(63.) | ![]() |
$F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ | $$F_1F_2=F_1\langle F _ 2\rangle =F_1(F_2)=F_2(F_1)=F_2\langle F _ 1\rangle,$$ | conf 0.628
e03696024.png (24) |
Formal group
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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43.(120.)* | ![]() |
$\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ | $$\operatorname {log}F_{\rm MU }(X)=\sum _ {i=1}^{\infty }i^{-1}[{\rm CP}^{i-1}]X^i,$$ | conf 0.098 F
f040820118.png (118) |
44.(147.)* | ![]() |
$( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ | $$(x_1,\ldots ,x_n)\circ (y_1,\ldots ,y_n)=(z_1,\ldots ,z_n),$$ | conf 0.553 F
f04082059.png (59) |
Gel'fond-Schneider method
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45.(148.) | ![]() |
$\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$ | $\alpha ^{\beta }=\operatorname {exp}\{\beta \operatorname {log}\alpha \}$ | conf 0.979
g1300205.png (5) |
Group
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46.(22.)* | ![]() |
$\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$ | source incomplete | conf 0.226 F
g04521075.png (75) |
Homogeneous space
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47.(89.) | ![]() |
$\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$ | $$\mathfrak g=\mathfrak f+\mathfrak m,\quad \mathfrak f\cap \mathfrak m=\{0\},$$ | conf 0.793
h04769069.png (69) |
Hopf algebra
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48.(103.) | ![]() |
$m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$ | $m\circ (\iota \otimes 1 )\circ \mu =m\circ (1\otimes \iota )\circ \mu =e\circ \epsilon$ | conf 0.618
h047970129.png (129) |
49.(107.)* | ![]() |
$F _ { 1 } ( X || Y ) , \ldots , F _ { n } ( X || Y ) \in K [ X _ { 1 } , \ldots , X _ { n } || Y _ { 1 } , \ldots , Y _ { n } ] \}$ | $F_1(X;Y),\ldots ,F_n(X;Y)\in K [X_1,\ldots ,X_n;Y_1,\ldots ,Y_n]\}$ | conf 0.353 F
h047970139.png (139) |
50.(97.) | ![]() |
$\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$ | $$\epsilon (x)=0,\quad \delta (x)=x\otimes 1 +1\otimes x ,\quad x \in \mathfrak g.$$ | conf 0.213
h04797042.png (42) |
Invariants, theory of
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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51.(149.)* | ![]() |
$\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ | $$\alpha _ {i_1,\dots,i_n}\rightarrow \alpha _ {i_1,\dots,i_n}^{\prime }.$$ | conf 0.142 F
i05235015.png (15) |
Jordan algebra
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52.(150.) | ![]() |
$H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ | $$(C_3,\Gamma )=\big\{X\in C _ 3:X=\Gamma ^{-1}X\square ^{\prime }\Gamma \big\},$$ | conf 0.651
j05427030.png (30) |
53.(42.) | ![]() |
$\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$ | $$\Gamma =\operatorname {diag}\{\gamma _ 1,\gamma _ 2,\gamma _ 3\},\quad \gamma _ i\neq 0 ,\quad \gamma _ i\in F,$$ | conf 0.987
j05427031.png (31) |
54.(125.)* | ![]() |
$\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$ | $\mathfrak g=\mathfrak g_{-1}+\mathfrak g_0+\mathfrak g_1$ | conf 0.598 F
j05427077.png (77) |
Jordan matrix
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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55.(6.)* | ![]() |
$J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$ | $$J=\left\| \begin {array}{cccc} J_{n_1}(\lambda_1) &0 &0 &0\\ 0 &\ddots &\ddots &0\\ 0 &\ddots &\ddots &0\\ 0 &0 &0 &J_{n_s}(\lambda_s) \end {array} \right\|,$$ | conf 0.072 F
j0543403.png (3) |
56.(64.) | ![]() |
$C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ | $$C_m(\lambda )=\operatorname {rk}(A-\lambda E )^{m-1}-2\operatorname {rk}(A-\lambda E )^m+$$ | conf 0.955
j05434030.png (30) |
57.(1.)* | ![]() |
$J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$ | $$J_m(\lambda)=\left\| \begin {array}{cccccc} \lambda &1 &\square &\square &\square &\square \\ \square &\lambda &1 &\square &0 &\square \\ \square &\square &\ddots &\ddots &\square &\square\\ \square &\square &\square &\ddots &\ddots &\square \\ \square &0 &\square &\square &\lambda &1\\ \square &\square &\square &\square &\square &\lambda \end {array} \right\|,$$ | conf 0.098 F
j0543406.png (6) |
Lie algebra, semi-simple
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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58.(5.) | ![]() |
$\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ | $$B_n:\quad \left\| \begin {array}{rrrrrr} 2 &{-1} &0 &{\dots } &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot \\ 0 &0 &0 &{\dots } &{-1} &0\\ 0 &0 &0 &{\dots } &2 &{-2}\\ 0 &0 &0 &{\dots } &{-1} &2 \end {array} \right\|,$$ | conf 0.232
l058510127.png (127) |
59.(3.)* | ![]() |
$\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ | $$D_n:\quad \left\| \begin {array}{rrrrrrr} 2 &{-1} &0 &{\dots } &0 &0 &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0 &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0 &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot &\cdot &\cdot \\ 0 &0 &0 &{\dots } &2 &{-1} &0 &0\\ 0 &0 &0 &{\dots } &{-1} &2 &{-1} &{-1}\\ 0 &0 &0 &{\dots } &0 &{-1} &2 &0\\ 0 &0 &0 &{\dots } &0 &{-1} &0 &2 \end {array} \right\|,$$ | conf 0.055 F
l058510129.png (129) |
60.(8.)* | ![]() |
$\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ | $$E_6: \quad \left\| \begin {array}{rrrrrr} 2 &0 &{-1} &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0\\ {-1} &0 &2 &{-1} &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0\\ 0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ | conf 0.628 F
l058510130.png (130) |
61.(4.) | ![]() |
$\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ | $$E_7:\quad \left\| \begin {array}{rrrrrrr} 2 &0 &{-1} &0 &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ | conf 0.278
l058510131.png (131) |
62.(2.)* | ![]() |
$\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ | $$E_8:\quad \left\| \begin {array}{rrrrrrrr} 2 &0 &{-1} &0 &0 &0 &0 & 0\\ 0 &2 &0 &{-1} &0 &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ | conf 0.354 F
l058510132.png (132) |
63.(10.)* | ![]() |
$\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$ | $$F_4:\quad \left\| \begin {array}{rrrr} 2 &{-1} &0 &0\\ {-1} &2 &{-2} &0\\ 0 &{-1} &2 &{-1}\\ 0 &0 &{-1} &2 \end {array} \right\|,\quad G _ 2:\quad \left\| \begin {array}{rr} 2&{-1}\\ {-3}&2 \end {array} \right\|.$$ | conf 0.374 F
l058510133.png (133) |
64.(98.) | ![]() |
$\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$ | $$\mathfrak g_{\alpha }=\{X\in \mathfrak g:[H,X]=\alpha (H)X,H\in \mathfrak h\}.$$ | conf 0.976
l05851030.png (30) |
65.(126.) | ![]() |
$\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ | $$\mathfrak g=\mathfrak h+\sum _ {\alpha \in \Sigma }\mathfrak g_{\alpha }.$$ | conf 0.945
l05851037.png (37) |
66.(61.)* | ![]() |
$\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$ | $$\mathfrak g_{\alpha }=\operatorname {dim}[\mathfrak g_{\alpha },\mathfrak g_{-\alpha }]=1.$$ | conf 0.520 F
l05851044.png (44) |
67.(65.)* | ![]() |
$[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ | $$[H_{\alpha },X_{\alpha }]=2X_{\alpha }\quad {\rm and }\quad [H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }.$$ | conf 0.539 F
l05851050.png (50) |
68.(70.) | ![]() |
$\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$ | $$\beta (H_{\alpha })=\frac {2(\alpha ,\beta )}{(\alpha ,\alpha )},\quad \alpha ,\beta \in \Sigma,$$ | conf 0.997
l05851051.png (51) |
69.(112.) | ![]() |
$[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ | $$[\mathfrak g_{\alpha },\mathfrak g_{\beta }]=\mathfrak g_{\alpha +\beta }$$ | conf 0.917
l05851057.png (57) |
70.(127.) | ![]() |
$H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$ | $$H_{\alpha _ 1},\ldots ,H_{\alpha _ k},X_{\alpha }\quad (\alpha \in \Sigma )$$ | conf 0.432
l05851064.png (64) |
71.(113.)* | ![]() |
$[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ | $$[[X_{\alpha _ i},X_{-}\alpha _ i],X_{-\alpha _ j}]=-n(i,j)X_{\alpha _ j},$$ | conf 0.628 F
l05851069.png (69) |
72.(79.) | ![]() |
$n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$ | $$n(i,j)=\alpha _ j(H_i)=\frac {2(\alpha _ i,\alpha _ j)}{(\alpha _ j,\alpha _ j)}.$$ | conf 0.992
l05851073.png (73) |
73.(13.) | ![]() |
$[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ | $$[X_{\alpha },X_{\beta }]=\left\{ \begin {array}{ll} {N_{\alpha ,\beta }X_{\alpha +\beta }} &{\text{ if }\alpha +\beta \in \Sigma,}\\ 0 &{\text{ if }\alpha +\beta \notin \Sigma,} \end {array} \right.$$ | conf 0.988
l05851074.png (74) |
74.(80.) | ![]() |
$N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$ | $$N_{\alpha ,\beta }=-N_{-\alpha ,-\beta }\quad {\rm and }\quad N _ {\alpha ,\beta }=\pm (p+1),$$ | conf 0.961
l05851078.png (78) |
75.(85.)* | ![]() |
$X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ | $$iH_\alpha,X_{\alpha }-X_{-\alpha },\quad i (X_{\alpha }+X_{-\alpha })\quad (\alpha \in \Sigma _ {+})$$ | conf 0.691 F
l05851085.png (85) |
Lie algebra, solvable
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
76.(119.)* | ![]() |
$[ \mathfrak { g } _ { i } , \mathfrak { g } _ { i } ] \subset \mathfrak { g } _ { \mathfrak { i } } + 1$ | $[\mathfrak g_i,\mathfrak g_i]\subset \mathfrak g_{i+1}$ | conf 0.276 F
l05852011.png (11) |
77.(141.) | ![]() |
$\operatorname { dim } \mathfrak { g } _ { i } = \operatorname { dim } \mathfrak { g } - i$ | $\operatorname {dim}\mathfrak g_i=\operatorname {dim}\mathfrak g-i$ | conf 0.901
l05852046.png (46) |
Lie group
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
78.(62.)* | ![]() |
$( G ) \cong \operatorname { Aut } ( L ( G ) ) \quad \text { and } \quad L ( \operatorname { Aut } ( G ) ) \cong D ( L ( G ) )$ | $$\operatorname {Aut}(G)\cong \operatorname {Aut}(L(G))\quad {\rm and }\quad L (\operatorname {Aut}(G))\cong D (L(G)),$$ | conf 0.693 F
l058590115.png (115) |
79.(50.) | ![]() |
$( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ | $$(X,Y)\rightarrow \operatorname {exp}^{-1}(\operatorname {exp}X\operatorname {exp}Y),\quad X ,Y\in L (G),$$ | conf 0.856
l05859086.png (86) |
Lie group, compact
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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80.(121.)* | ![]() |
$J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$ | $$J=\left\| \begin {array}{cc} 0 &{E_x}\\ {-E_x} &0 \end {array} \right\|,$$ | conf 0.364 F
l05861012.png (12) |
Lie group, nilpotent
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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81.(83.) | ![]() |
$N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$ | $$N(F)=\{g\in GL (V):gv\equiv v \operatorname {mod}V_i\;\text{for all }v\in V _ i,\;i\geq 1 \}$$ | conf 0.466
l0586604.png (4) |
Lie group, semi-simple
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
82.(35.)* | ![]() |
$L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$ | $$L(\mathfrak g)\cong \Gamma _ 0(\mathfrak u)\cap \mathfrak h^{\prime }/\Gamma _ 0([\mathfrak k,\mathfrak k])$$ | conf 0.659 F
l058680102.png (102) |
83.(81.)* | ![]() |
$\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ | $$\Gamma _ 1=\Gamma _ 1(g)=\{X\in h :\alpha (X)\in 2 \pi i {\mathbf Z}\;\text{for all }\alpha \in \Sigma \}.$$ | conf 0.183 F
l05868032.png (32) |
Lie p-algebra
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
84.(36.) | ![]() |
$( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$ | $$(\operatorname {ad}x)^ny=\sum _ {j=1}^n(-1)^j\left(\begin {array}cn\\ j \end {array} \right)x^{n-j}yx^j$$ | conf 0.356
l05872026.png (26) |
85.(99.) | ![]() |
$\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ | $$\pi (x+y)=\pi (x)+\pi (y),\quad \pi (\lambda x )=\lambda ^p\pi (x),\quad \lambda \in k .$$ | conf 0.964
l05872078.png (78) |
Lie theorem
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
86.(134.) | ![]() |
$y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ | $$y_i=f_i(g_1,\ldots ,g_i;x_1,\ldots ,x_n),\quad i =1,\ldots ,n$$ | conf 0.276
l05876010.png (10) |
87.(86.) | ![]() |
$X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ | $$X_i=\sum _ {j=1}^n\xi _ {ij}(x)\frac {\partial }{\partial x _ j},\quad i =1,\ldots ,r,$$ | conf 0.656
l05876016.png (16) |
88.(66.)* | ![]() |
$\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$ | $$\frac {\partial f _ j}{\partial g _ i}(g,x)=\sum _ {k=1}^r\xi _ {kj}(f(g_sx))\psi _ {ki}(g),$$ | conf 0.336 F
l05876030.png (30) |
89.(19.)* | ![]() |
$\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ | $$\sum _ {k=1}^N(\xi _ {ik}\frac {\partial \xi _ {jl}}{\partial x _ k}-\xi _ {jk}\frac {\partial \xi _ {il}}{\partial x _ k})=\sum _ {k=1}^rc_{ij}^k\xi _ {kl},$$ | conf 0.157 F
l05876037.png (37) |
90.(14.) | ![]() |
$\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ | $$\left.\begin {array}c{c_{ij}^k=-c_{ji}^k},\\ {\displaystyle\sum _ {l=1}^r(c_{il}^mc_{jk}^l+c_{kl}^mc_{ij}^l+c_{jl}^mc_{ki}^l)=0,\quad 1 \leq i ,j,k,l,m\leq r,} \end {array} \right\}$$ | conf 0.085
l05876052.png (52) |
Maximal torus
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
91.(95.) | ![]() |
$F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$ | $$F(x_1f_1+\ldots +x_xf_n)=x_1x_n+x_2x_{n-1}+\ldots +x_px_{n-p+1},$$ | conf 0.198
m06301072.png (72) |
Non-Abelian cohomology
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
92.(114.)* | ![]() |
$\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ | $$\phi (g_1)\phi (g_2)\phi (g_1g_2)^{-1}=\operatorname {Int}m(g_1,g_2),$$ | conf 0.443 F
n066900110.png (110) |
93.(90.)* | ![]() |
$( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$ | $$m'(g_1,g_2)=h(g_1)(\phi (g_1)(h(g_2)))m(g_1,g_2)h(g_1,g_2)^{-1}.$$ | conf 0.764 F
n066900118.png (118) |
94.(44.) | ![]() |
$\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ | $$\delta (e)=e\quad \;\text{and }\quad \delta (\rho (a)b)=\sigma (a)\delta (b),\quad \alpha \in C ^0,\quad b \in C ^1,$$ | conf 0.400
n06690016.png (16) |
95.(60.)* | ![]() |
$C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$ | $$C^{*}(\mathfrak U,{\cal F})=(C^0(\mathfrak U,{\cal F}),C^1(\mathfrak U,{\cal F}),C^2(\mathfrak U,{\cal F})),$$ | conf 0.205 F
n06690028.png (28) |
Picard scheme
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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96.(39.)* | ![]() |
$\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$ | $$\operatorname {Pic}_{X/k}(S^{\prime })=\operatorname {Pic}(X\times_k S ^{\prime })/\operatorname {Pic}(S^{\prime })$$ | conf 0.345 F +
p07267025.png (25) |
Principal analytic fibration
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
97.(100.)* | ![]() |
$g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$ | $$g_j:U_i\cap U _ j\rightarrow G ,\quad i ,j\in I ,\quad U _ i\cap U _ j\neq \emptyset,$$ | conf 0.184 F
p07464025.png (25) |
Quantum groups
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
98.(101.) | ![]() |
$\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$ | $$\phi ^{*}:\mathfrak g^{*}\otimes \mathfrak g^{*}\rightarrow \mathfrak g^{*}$$ | conf 0.837
q07631062.png (62) |
99.(108.) | ![]() |
$\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ | $$\delta :U_{\mathfrak g}\rightarrow U _ {\mathfrak g}\otimes U _ {\mathfrak g}$$ | conf 0.648
q07631071.png (71) |
100.(56.)* | ![]() |
$\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ | $$\delta (\alpha )=\operatorname {lim}_{h\rightarrow 0 }h^{-1}(\Delta (a)-\Delta ^{\prime }(\alpha ))$$ | conf 0.304 F
q07631072.png (72) |
101.(129.)* | ![]() |
$[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$ | $$[\alpha ,X_i^{\pm }]=\pm \alpha _ i(\alpha )X_i^{\pm }\quad \text{for }a\in \mathfrak h;$$ | conf 0.544 F
q07631088.png (88) |
102.(128.) | ![]() |
$[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ | $$[X_i^{+},X_j^{-}]=2\delta _ {ij}h^{-1}\operatorname {sinh}(hH_i/2).$$ | conf 0.893
q07631089.png (89) |
103.(20.) | ![]() |
$\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$ | $$\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)q^{-k(n-k)/2}(X_i^{\pm })^kX_j^{\pm }\cdot (X_i^{\pm })^{n-k}=0.$$ | conf 0.055
q07631092.png (92) |
104.(30.) | ![]() |
$\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ | $$\left( \begin {array}ln\\ k \end {array} \right)_q=\frac {(q^n-1)\ldots (q^{n-k+1}-1)}{(q^k-1)\ldots (q-1)} .$$ | conf 0.443
q07631095.png (95) |
105.(21.)* | ![]() |
$\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$ | $$\Delta (X_i^{\pm })=X_i^{\pm }\otimes \operatorname {exp}(\frac {hH_i}4)+\operatorname {exp}(\frac {-hH_i}4)\otimes X _ i^{\pm }.$$ | conf 0.212 F
q07631099.png (99) |
Rational representation
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
106.(91.) | ![]() |
$0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$ | $$0\leq \frac {2(\chi ,\alpha )}{(\alpha ,\alpha )}<p\quad \text{for all }\alpha \in \Delta.$$ | conf 0.879
r077630100.png (100) |
107.(135.) | ![]() |
$\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ | $$\phi _ 0\otimes \phi _ 1^{Fr}\otimes \ldots \otimes \phi _ d^{{Fr}^d},$$ | conf 0.136
r077630104.png (104) |
108.(45.)* | ![]() |
$\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$ | $$\chi =\delta _ {\phi }-\sum _ {\alpha \in \Delta }m_{\alpha }\alpha ,\quad m _ {\alpha }\in Z ,\quad m _ {\alpha }\geq 0.$$ | conf 0.862 F
r07763055.png (55) |
Singular point
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
109.(31.) | ![]() |
$\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ | $$\sum _ {k_1,\ldots ,k_n=0}^{\infty }c_{k_1\cdots k _ n}(z_1-\zeta _ 1)^{k_1}\ldots (z_n-\zeta _ n)^{k_n}$$ | conf 0.324
s085590225.png (225) |
110.(46.) | ![]() |
$\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ | $$\frac {m_1}{n_1}<\frac {m_2}{n_1n_2}<\ldots <\frac {m_g}{n_1\ldots n _ g}=\frac {m_g}n$$ | conf 0.459
s085590404.png (404) |
111.(115.)* | ![]() |
$p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ | $$p(Z)=1-\operatorname {dim}H^0({\mathbf Z},{\cal O}_{\mathbf Z })+\operatorname {dim}H^1({\mathbf Z},{\cal O}_{\mathbf Z })$$ | conf 0.997 F
s085590429.png (429) |
112.(136.)* | ![]() |
$X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$ | $$X_{\epsilon }=\{(x_0,\ldots ,x_x):f(x_0,\ldots ,x_x)=\epsilon \}$$ | conf 0.433 F
s085590440.png (440) |
113.(12.) | ![]() |
$= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ | $$=\left\{ \begin {array}{ll} {(x+\lambda )^2\ldots (x+k\lambda )^2} &{\text{ if }\mu =2k,}\\ {(x+\lambda )^2\ldots (x+k\lambda )^2(x+(k+1)\lambda )} &{\text{ if }\mu =2k+1,} \end {array} \right.$$ | conf 0.870
s085590458.png (458) |
114.(75.) | ![]() |
$( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ | $$\big(\frac {\partial F (x,y,\lambda )}{\partial x },\frac {\partial F (x,y,\lambda )}{\partial y }\big)$$ | conf 0.986
s085590482.png (482) |
115.(137.) | ![]() |
$\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$ | $$\frac {dx_i}{dx_{i_0}}=f_i(x),\quad f _ i\in C (U),\quad i \neq i _ 0.$$ | conf 0.594
s085590515.png (515) |
116.(142.)* | ![]() |
$A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ | $$A=\left\| \begin {array}{ll} {\alpha } &b\\ c &e \end {array} \right\|$$ | conf 0.506 F
s085590527.png (527) |
117.(53.) | ![]() |
$\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$ | $$\Delta =(F_{xx}^{\prime \prime })_0(F_{yy}^{\prime \prime })_0-(F_{xy}^{\prime \prime })_0^2$$ | conf 0.920
s085590634.png (634) |
118.(16.)* | ![]() |
$\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ | $$\left\| \begin {array}{lll} {F_x^{\prime }} &{F_y^{\prime }} &{F_z^{\prime }}\\ {G_x^{\prime }} &{G_y^{\prime }} &{G_Z^{\prime }} \end {array} \right\|$$ | conf 0.230 F
s085590645.png (645) |
119.(92.) | ![]() |
$( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$ | $$(F_x^{\prime })_0=0,\quad (F_y^{\prime })_0=0,\quad (F_z^{\prime })_0=0.$$ | conf 0.300
s085590653.png (653) |
Solv manifold
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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120.(138.) | ![]() |
$\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$ | $$\{e\}\rightarrow \Delta \rightarrow \pi \rightarrow {\mathbf Z}^s\rightarrow \{e\}$$ | conf 0.972
s08610054.png (54) |
Stability theorems in algebraic K-theory
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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121.(71.) | ![]() |
$\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$ | $$\psi _ {t_1,\ldots ,t_n}^{\prime }:SK_1(R)\rightarrow S K _ 1(R(t_1,\ldots ,t_n)).$$ | conf 0.379
s08706033.png (33) |
Steinberg module
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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122.(130.) | ![]() |
$e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$ | $$e=\frac {|U|}{|G|}\big(\sum _ {b\in B }b\big)\big(\sum _ {w\in W }\operatorname {sign}(w)w\big)$$ | conf 0.138
s13053016.png (16) |
Steinberg symbol
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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123.(24.)* | ![]() |
$( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ | $$(x_{ij}(a),x_{kl}(b))=\left\{ \begin {array}{ll} 1 &{\text{ if }i\neq l ,j\neq k },\\ {x_{il}(ab)} &{\text{ if }i\neq l ,j=k}. \end {array} \right.$$ | conf 0.381 F
s13054017.png (17) |
Tilting theory
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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124.(84.) | ![]() |
$0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$ | $$0\rightarrow \Lambda \rightarrow T _ 1\rightarrow \ldots \rightarrow T _ n\rightarrow 0 $$ | conf 0.946
t130130105.png (105) |
Tits quadratic form
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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125.(18.) | ![]() |
$q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ | $$q_R(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}x_ix_j+\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}r_{i,j}x_ix_j,$$ | conf 0.112
t130140104.png (104) |
126.(40.) | ![]() |
$[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$ | $$[X]\mapsto \chi _ R([X])=\sum _ {m=0}^{\infty }(-1)^m\operatorname {dim}_K\operatorname {Ext}_R^m(X,X)$$ | conf 0.116
t130140118.png (118) |
127.(132.)* | ![]() |
$\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ | $$\underline {\dim }:K_0(\operatorname {mod}R)\rightarrow {\mathbf Z}^{Q_0}$$ | conf 0.287 F
t130140119.png (119) |
128.(37.)* | ![]() |
$q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ | $$q_I(x)=\sum _ {i\in I }x_i^2+\sum _ {i\prec j \atop j\in I\setminus {\rm max}I}x_ix_j-\sum _ {p\in \operatorname {max}I}\big(\sum _ {i\prec p }x_i\big)x_p$$ | conf 0.197 F
t130140140.png (140) |
129.(131.)* | ![]() |
$X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ | $$X\mapsto \underline {\dim }X=(\operatorname {dim}_KX_j)_{j\in Q _ 0}$$ | conf 0.819 F
t13014044.png (44) |
130.(25. | ![]() |
$[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$ | $$[X]\mapsto \chi _ Q([X])=\operatorname {dim}_K\operatorname {End}_Q(X)-\operatorname {dim}_K\operatorname {Ext}_Q^1(X,X)$$ | conf 0.661
t13014048.png (48) |
131.(38.)* | ![]() |
$A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ | $$A_Q(v)=\prod _ {i,j\in Q _ 0}\prod _ {\langle \beta :j\rightarrow i \rangle \in Q _ 1}M_{v_i\times v _ j}(K)_{\beta }$$ | conf 0.481 F
t13014056.png (56) |
132.(139.)* | ![]() |
$\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ | $$q_Q(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {i,j\in Q _ 0}d_{ij}x_ix_j,$$ | conf 0.648 F
t1301406.png (6) |
Torus
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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133.(41.)* | ![]() |
$r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ | $$r=\alpha \operatorname {sin}u{\bf k}+l(1+\epsilon \operatorname {cos}u)({\bf i}\operatorname {cos}v+{\bf j}\operatorname {sin}v)$$ | conf 0.585 F
t0933502.png (2) |
134.(122.)* | ![]() |
$d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$ | $$ds^2=\alpha ^2du^2+l^2(1+\epsilon \operatorname {cos}u)^2dv^2,$$ | conf 0.696 F
t0933507.png (7) |
Uniform distribution
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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135.(9.) | ![]() |
$u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ | $$u_3(x)=\left\{ \begin {array}{ll} {\frac {x^2}2,} &{0\leq x <1,}\\ {\frac {[x^2-3(x-1)^2]}2,} &{1\leq x <2,}\\ {\frac {[x^2-3(x-1)^2+3(x-2)^2]}2,} &{2\leq x <3,}\\ {0,} &{x\notin [0,3].} \end {array} \right.$$ | conf 0.733
u09524027.png (27) |
136.(32.)* | ![]() |
$p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$ | $$p(x)=\left\{ \begin {array}{ll} {\frac 1{b-\alpha },} &{x\in [\alpha ,b],}\\ {0,} &{x\notin [\alpha ,b].} \end {array} \right.$$ | conf 0.681 F
u0952403.png (3) |
137.(34.) | ![]() |
$u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ | $$u_n(x)=\frac 1{(n-1)!}\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)(x-k)_{+}^{n-1}$$ | conf 0.569
u09524030.png (30) |
138.(109.) | ![]() |
$z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$ | $$z_{+}=\left\{ \begin {array}{ll} {z,} &{z>0}.\\ {0,} &{z\leq 0 }. \end {array} \right.$$ | conf 0.676
u09524034.png (34) |
139.(43.) | ![]() |
$F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ | $$F(x)=\left\{ \begin {array}{ll} {0,} &{x\leq a },\\ {\frac {x-a}{b-a},} &{a<x\leq b },\\ {1,} &{x>b}, \end {array} \right.$$ | conf 0.468
u0952407.png (7) |
140.(47.) | ![]() |
$p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$ | $$p(x_1,\ldots ,x_n)=\left\{ \begin {array}{ll} {C\neq 0 ,} &{x\in D },\\ {0,} &{x\notin D }, \end {array} \right.$$ | conf 0.705
u09524072.png (72) |
Unipotent group
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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141.(143.) | ![]() |
$\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$ | $$\{g\in \operatorname {GL}(V):(1-g)^n=0\},\quad n =\operatorname {dim}V,$$ | conf 0.287
u0954106.png (6) |
Weyl module
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
142.(51.) | ![]() |
$\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$ | $$\operatorname {diag}(t_1,\ldots ,t_n)\mapsto t _ 1^{\lambda _ 1}\ldots t _ n^{\lambda _ n}\in K,$$ | conf 0.507
w120090122.png (122) |
143.(54.)* | ![]() |
$\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ | $$\chi _ {\lambda }=\sum _ {\mu \in \Lambda (n)}\operatorname {dim}_K(\Delta (\lambda )^{\mu })_{e_{\mu }},$$ | conf 0.461 F
w120090135.png (135) |
144.(110.) | ![]() |
$\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$ | $$\mathfrak B=\{e_{\pm }\alpha ,h_{\beta }:\alpha \in \Phi ^{+},\beta \in \Sigma \}.$$ | conf 0.381
w120090259.png (259) |
145.(82.) | ![]() |
$\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ | $$\left( \begin {array}ch\\ i \end {array} \right)=\frac {h(h-1)\ldots (h-i+1)}{i!} $$ | conf 0.487
w120090342.png (342) |
146.(28.)* | ![]() |
$\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$ | $$\mathfrak S_{\{1,\ldots ,\lambda _ 1\}}\times \mathfrak S_{\{\lambda _ 1+1,\ldots ,\lambda _ 1+\lambda _ 2\}}\times \dots $$ | conf 0.312 F
w12009095.png (95) |
147.(104.) | ![]() |
$\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ | $$\ldots \times \mathfrak S_{\{\lambda _ 1+\ldots +\lambda _ {n-1}+1,\ldots ,r\}},$$ | conf 0.259
w12009096.png (96) |
Witt vector
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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148.(87.)* | ![]() |
$\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ | $$\langle \alpha ><b\rangle =\langle \alpha b \rangle ,\quad \langle {\bf 1}\rangle ={\bf f}_1={\bf V}_1=\text{ unit element}1,$$ | conf 0.351 F
w098100172.png (172) |
149.(123.)* | ![]() |
$\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ | $$\langle \alpha +b\rangle =\sum _ {n=1}^{\infty }{\bf V}_n\langle r _ n(\alpha ,b){\bf f}_n.$$ | conf 0.143 F
w098100177.png (177) |
150.(102.) | ![]() |
$\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$ | $$\sigma (\alpha _ 1,\alpha _ 2,\ldots )=(\alpha _ 1^p,\alpha _ 2^p,\ldots )$$ | conf 0.771
w098100190.png (190) |
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=44207